{"id":4548,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/"},"modified":"2018-05-04T14:59:12","modified_gmt":"2018-05-04T09:29:12","slug":"ncert-solutions-class-12-maths-exercise-6-5","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 6.5"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\" >NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#2_Find_the_maximum_and_minimum_values_if_any_of_the_following_functions_given_by\" >2. Find the maximum and minimum values, if any, of the following functions given by:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-2\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#3_Find_the_local_maxima_and_local_minima_if_any_of_the_following_functions_Find_also_the_local_maximum_and_the_local_minimum_values_as_the_case_may_be\" >3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-3\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#4_Prove_that_the_following_functions_do_not_have_maxima_or_minima\" >4. Prove that the following functions do not have maxima or minima:<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#But_this_is_not_possible_Therefore_there_is_no_turning_point\" >But this is not possible. Therefore, there is no turning point.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#5_Find_the_absolute_maximum_value_and_the_absolute_minimum_value_of_the_following_functions_in_the_given_intervals\" >5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#6_Find_the_maximum_profit_that_a_company_can_make_if_the_profit_function_is_given_by\" >6. \u00a0Find the maximum profit that a company can make, if the profit function is given by \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#9_What_is_the_maximum_value_of_the_function\" >9. What is the maximum value of the function \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#10_Find_the_maximum_value_of_in_the_interval_1_3_Find_the_maximum_value_of_the_same_function_in\" >10. Find the maximum value of \u00a0in the interval [1, 3]. Find the maximum value of the same function in \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#11_It_is_given_that_at_the_function_attains_its_maximum_value_on_the_interval_0_20_Find_the_value_of\" >11. It is given that at \u00a0the function \u00a0attains its maximum value, on the interval [0, 20]. Find the value of \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-4\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#12_Find_the_maximum_and_minimum_value_of_on\" >12. Find the maximum and minimum value of \u00a0on \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-5\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#13_Find_two_numbers_whose_sum_is_24_and_whose_product_is_as_large_as_possible\" >13. Find two numbers whose sum is 24 and whose product is as large as possible.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-6\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#14_Find_two_positive_integers_and_such_that_and_is_maximum\" >14. Find two positive integers \u00a0and \u00a0such that \u00a0and \u00a0is maximum.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-7\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#15_Find_two_positive_integers_and_such_that_their_sum_is_35_and_the_product_is_a_maximum\" >15. Find two positive integers \u00a0and \u00a0such that their sum is 35 and the product \u00a0is a maximum.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-8\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#16_Find_two_positive_integers_whose_sum_is_16_and_sum_of_whose_cubes_is_minimum\" >16. Find two positive integers whose sum is 16 and sum of whose cubes is minimum.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-9\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#17_A_square_piece_of_tin_of_side_18_cm_is_to_be_made_into_a_box_without_top_by_cutting_a_square_from_each_corner_and_folding_up_the_flaps_to_form_the_box_What_should_be_the_side_of_the_square_to_be_cut_off_so_that_the_volume_of_the_box_is_the_maximum_possible\" >17. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-10\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#18_A_rectangular_sheet_of_tin_45_cm_by_24_cm_is_to_be_made_into_a_box_without_top_by_cutting_off_square_from_each_corner_and_folding_up_the_flaps_What_should_be_the_side_of_the_square_to_be_cut_off_so_that_the_volume_of_the_box_is_maximum\" >18. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-11\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#19_Show_that_of_all_the_rectangles_inscribed_in_a_given_fixed_circle_the_square_has_maximum_area\" >19. Show that of all the rectangles inscribed in a given fixed circle, the square has maximum area.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-12\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#20_Show_that_the_right_circular_cylinder_of_given_surface_and_maximum_volume_is_such_that_its_height_is_equal_to_the_diameter_of_the_base\" >20. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-13\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#21_Of_all_the_closed_cylindrical_cans_right_circular_of_a_given_volume_of_100_cubic_centimeters_find_the_dimensions_of_the_can_which_has_the_minimum_surface_area\" >21. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimeters, find the dimensions of the can which has the minimum surface area.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-14\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#22_A_wire_of_length_28_m_is_to_be_cut_into_two_pieces_One_of_the_pieces_is_to_be_made_into_a_square_and_the_other_into_a_circle_What_should_be_the_length_of_the_two_pieces_so_that_the_combined_area_of_the_square_and_the_circle_is_minimum\" >22. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-15\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#23_Prove_that_the_volume_of_the_largest_cone_that_can_be_inscribed_in_a_sphere_of_radius_R_is_of_the_volume_of_the_sphere\" >23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \u00a0of the volume of the sphere.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-16\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#24_Show_that_the_right_circular_cone_of_least_curve_surface_and_given_volume_has_an_altitude_equal_to_time_the_radius_of_the_base\" >24. Show that the right circular cone of least curve surface and given volume has an altitude equal to \u00a0time the radius of the base.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-17\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#25_Show_that_the_semi-vertical_angle_of_the_cone_of_the_maximum_value_and_of_given_slant_height_is\" >25. Show that the semi-vertical angle of the cone of the maximum value and of given slant height is \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-18\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#26_Show_that_the_semi-vertical_angle_of_the_right_circular_cone_of_given_surface_area_and_maximum_volume_is\" >26. Show that the semi-vertical angle of the right circular cone of given surface area and maximum volume is \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-19\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#27_The_point_on_the_curve_which_is_nearest_to_the_point_0_5_is\" >27. The point on the curve \u00a0which is nearest to the point (0, 5) is:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-46\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#NCERT_Solutions_class_12_Maths_Exercise_65-20\" >NCERT Solutions class 12 Maths Exercise 6.5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 6.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and\u00a0Maths \u00a0part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-application-of-derivatives\/1290\/ncert-solutions\/5\/\">NCERT solutions for Applications of Derivatives<\/a>\u00a0as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 6.5\" width=\"187\" height=\"243\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\"><\/span><strong>NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. Find the maximum and minimum values, if any, of the following functions given by:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 29px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image121.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image122.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 29px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image123.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image124.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 2177\" style=\"height: 29px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 2178\" style=\"height: 29px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image125.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 2179\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Adding 3 both sides,\u00a0 <img decoding=\"async\" id=\"Picture 2180\" style=\"height: 29px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image127.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2181\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 2182\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image129.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the minimum value of <img decoding=\"async\" id=\"Picture 2183\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is 3 when <img decoding=\"async\" id=\"Picture 2184\" style=\"height: 19px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image131.png\" \/>, i.e., <img decoding=\"async\" id=\"Picture 2185\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image132.png\" \/><\/p>\n<p style=\"text-align: justify;\">This function does not have a maximum value.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 2186\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image122.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2187\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2188\" style=\"height: 45px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image133.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2189\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2190\" style=\"height: 53px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image134.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 2191\" style=\"height: 53px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image135.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2192\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2193\" style=\"height: 49px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image136.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 2194\" style=\"height: 49px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image137.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 2195\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Subtracting 2 from both sides, <img decoding=\"async\" id=\"Picture 2196\" style=\"height: 49px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image138.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2197\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2198\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image139.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, minimum value of <img decoding=\"async\" id=\"Picture 2199\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 2200\" style=\"height: 17px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image140.png\" \/>\u00a0and is obtained when <img decoding=\"async\" id=\"Picture 2201\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image141.png\" \/>, i.e., <img decoding=\"async\" id=\"Picture 2202\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image142.png\" \/><\/p>\n<p style=\"text-align: justify;\">And this function does not have a maximum value.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 2203\" style=\"height: 29px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image143.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 2204\" style=\"height: 29px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image144.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 2205\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Multiplying both sides by <img decoding=\"async\" id=\"Picture 2206\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image145.png\" \/>\u00a0and adding 10 both sides,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2207\" style=\"height: 29px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image146.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2208\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 2209\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image147.png\" \/>\u00a0\u00a0 [Using eq. (i)]\n<p style=\"text-align: justify;\">Therefore, maximum value of <img decoding=\"async\" id=\"Picture 2210\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is 10 which is obtained when <img decoding=\"async\" id=\"Picture 2211\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image148.png\" \/>\u00a0i.e., <img decoding=\"async\" id=\"Picture 2212\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/><\/p>\n<p style=\"text-align: justify;\">And therefore, minimum value of <img decoding=\"async\" id=\"Picture 2213\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0does not exist.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 2214\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image124.png\" \/><\/p>\n<p style=\"text-align: justify;\">As <img decoding=\"async\" id=\"Picture 2215\" style=\"height: 15px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image150.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5170\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image151.png\" \/><\/p>\n<p style=\"text-align: justify;\">As <img decoding=\"async\" id=\"Picture 5169\" style=\"height: 15px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image152.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5168\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image153.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, maximum value and minimum value of <img decoding=\"async\" id=\"Picture 5167\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image154.png\" \/>\u00a0do not exist.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_Find_the_maximum_and_minimum_values_if_any_of_the_following_functions_given_by\"><\/span><strong>2. Find the maximum and minimum values, if any, of the following functions given by:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image155.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 27px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image156.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 27px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image157.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image158.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <img decoding=\"async\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image159.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5166\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image155.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 5165\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image160.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5164\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Subtracting 1 from both sides, <img decoding=\"async\" id=\"Picture 5163\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image161.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5162\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5161\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image162.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, minimum value of <img decoding=\"async\" id=\"Picture 5160\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 5159\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image145.png\" \/>\u00a0which is obtained when <img decoding=\"async\" id=\"Picture 5158\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image163.png\" \/>\u00a0i.e., <img decoding=\"async\" id=\"Picture 5157\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image164.png\" \/><\/p>\n<p style=\"text-align: justify;\">From eq. (i), maximum value of <img decoding=\"async\" id=\"Picture 5156\" style=\"height: 27px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image165.png\" \/>\u00a0hence it does not exist.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5155\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image166.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 5154\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image167.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5153\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Multiplying by <img decoding=\"async\" id=\"Picture 5152\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image145.png\" \/>\u00a0both sides and adding 3 both sides,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5151\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image168.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5150\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5149\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image169.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, maximum value of <img decoding=\"async\" id=\"Picture 5148\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image154.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 5147\" style=\"height: 19px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image170.png\" \/>\u00a0which is obtained when <img decoding=\"async\" id=\"Picture 5146\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image171.png\" \/>\u00a0i.e., <img decoding=\"async\" id=\"Picture 5145\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/><\/p>\n<p style=\"text-align: justify;\">From eq. (i), minimum value of <img decoding=\"async\" id=\"Picture 5144\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image153.png\" \/>\u00a0hence it does not exist.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5143\" style=\"height: 27px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image157.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Since\u00a0 <img decoding=\"async\" id=\"Picture 5142\" style=\"height: 19px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image173.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5141\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Adding 5 to all sides,\u00a0 <img decoding=\"async\" id=\"Picture 5140\" style=\"height: 19px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image174.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5139\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5138\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image175.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, minimum value of <img decoding=\"async\" id=\"Picture 5137\" style=\"height: 27px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image176.png\" \/>\u00a0is 4 and maximum value is 6.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5136\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image158.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0 <img decoding=\"async\" id=\"Picture 5135\" style=\"height: 19px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image177.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5134\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image126.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">Adding 3 to all sides,\u00a0 <img decoding=\"async\" id=\"Picture 5133\" style=\"height: 19px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image178.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5132\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5131\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image179.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, minimum value of <img decoding=\"async\" id=\"Picture 5130\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is 2 and maximum value is 4.<\/p>\n<p style=\"text-align: justify;\"><strong>(v)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5129\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image159.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Since\u00a0 <img decoding=\"async\" id=\"Picture 5128\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\">Adding 1 to both sides, <img decoding=\"async\" id=\"Picture 5127\" style=\"height: 19px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image181.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5126\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5125\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image182.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, neither minimum value not maximum value of <img decoding=\"async\" id=\"Picture 5124\" style=\"height: 27px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image176.png\" \/>\u00a0exists.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-2\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Find_the_local_maxima_and_local_minima_if_any_of_the_following_functions_Find_also_the_local_maximum_and_the_local_minimum_values_as_the_case_may_be\"><\/span><strong>3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image183.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image184.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 41px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image185.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <img decoding=\"async\" style=\"height: 27px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image186.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <img decoding=\"async\" style=\"height: 27px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image187.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(vi) <img decoding=\"async\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image188.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(vii) <img decoding=\"async\" style=\"height: 41px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image189.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(viii) <img decoding=\"async\" style=\"height: 28px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image190.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5123\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image183.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5122\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5121\" style=\"height: 27px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image192.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5120\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image193.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5119\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5118\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5117\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0\u00a0\u00a0 [Turning point]\n<p style=\"text-align: justify;\">Again, when <img decoding=\"async\" id=\"Picture 5116\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>, <img decoding=\"async\" id=\"Picture 5115\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image193.png\" \/>\u00a0\u00a0\u00a0[Positive]\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5114\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0is a point of local minima and local minimum value = <img decoding=\"async\" id=\"Picture 5113\" style=\"height: 29px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image196.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5112\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image184.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5111\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5110\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image197.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5109\" style=\"height: 27px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image198.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5108\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5107\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5106\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image200.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5105\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5104\" style=\"height: 29px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image201.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5103\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5102\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image202.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5101\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5100\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5099\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0[Turning points]\n<p style=\"text-align: justify;\">Again, when <img decoding=\"async\" id=\"Picture 5098\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/>,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5097\" style=\"height: 27px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image203.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5096\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5095\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/>\u00a0is a point of local maxima and local maximum value <img decoding=\"async\" id=\"Picture 5094\" style=\"height: 29px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image204.png\" \/><\/p>\n<p style=\"text-align: justify;\">And when <img decoding=\"async\" id=\"Picture 5093\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5092\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image205.png\" \/>\u00a0\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5091\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5090\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0is a point of local minima and local minimum value <img decoding=\"async\" id=\"Picture 5089\" style=\"height: 29px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image206.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5088\" style=\"height: 27px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image207.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5087\" style=\"height: 45px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image208.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5086\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5085\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image209.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5084\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image210.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5083\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5082\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5081\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5080\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5079\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image213.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5078\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5077\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image214.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5076\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5075\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image215.png\" \/>\u00a0\u00a0 [Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5074\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5073\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0can have values in both I and III quadrant.<\/p>\n<p style=\"text-align: justify;\">But, <img decoding=\"async\" id=\"Picture 5072\" style=\"height: 45px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image208.png\" \/>\u00a0therefore, <img decoding=\"async\" id=\"Picture 5071\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0is only in I quadrant.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5070\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5069\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image215.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5068\" style=\"height: 41px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image217.png\" \/>\u00a0[Turning point]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5067\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image218.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5066\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image210.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5065\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5064\" style=\"height: 27px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image219.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5063\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image220.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5062\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image221.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5061\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image222.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5060\" style=\"height: 23px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image223.png\" \/>\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5059\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5058\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image218.png\" \/>\u00a0is a point of local maxima and local maximum value<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5057\" style=\"height: 45px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image224.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5056\" style=\"height: 44px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image225.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5055\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image226.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5054\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image227.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5053\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5052\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image228.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5051\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image229.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5050\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5049\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5048\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image230.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5047\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5046\" style=\"height: 19px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image231.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5045\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5044\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image232.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5043\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5042\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image233.png\" \/>\u00a0\u00a0 [Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5041\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5040\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0can have values in both II and IV quadrant.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5039\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5038\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image233.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5037\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image234.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5036\" style=\"height: 45px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image235.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5035\" style=\"height: 45px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image236.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5034\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5033\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image237.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5032\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image238.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5031\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5030\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image239.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5029\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image240.png\" \/>\u00a0\u00a0\u00a0[Turning point]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5028\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image239.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5027\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image229.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5026\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image241.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5025\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5024\" style=\"height: 27px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image219.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5023\" style=\"height: 45px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image242.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5022\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image220.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5021\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image221.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5020\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image222.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5019\" style=\"height: 23px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image223.png\" \/>\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5018\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5017\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image239.png\" \/>\u00a0is a point of local maxima and local maximum value = <img decoding=\"async\" id=\"Picture 5016\" style=\"height: 45px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image243.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5015\" style=\"height: 45px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image244.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5014\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image245.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5013\" style=\"height: 44px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image225.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5012\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image240.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5011\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image229.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5010\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image246.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5009\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5008\" style=\"height: 27px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image219.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5007\" style=\"height: 45px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image247.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5006\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image245.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5005\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image248.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5004\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image249.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5003\" style=\"height: 23px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image250.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5002\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5001\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image240.png\" \/>\u00a0is a point of local maxima and local maximum value = <img decoding=\"async\" id=\"Picture 5000\" style=\"height: 45px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image251.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4999\" style=\"height: 45px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image252.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4998\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image220.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 4997\" style=\"height: 44px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image253.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(v)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4996\" style=\"height: 27px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image187.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4995\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4994\" style=\"height: 27px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image254.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 4993\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image255.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4992\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4927\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4926\" style=\"height: 21px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image256.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4925\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4924\" style=\"height: 21px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image257.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4923\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4922\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image258.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4921\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4920\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 4919\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/>\u00a0\u00a0\u00a0[Turning points]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4918\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image260.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4916\" style=\"height: 27px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image261.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4915\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4914\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0is a point of local maxima and local maximum value is <img decoding=\"async\" id=\"Picture 4913\" style=\"height: 29px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image262.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4912\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image263.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4911\" style=\"height: 27px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image264.png\" \/>\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4910\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4909\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/>\u00a0is a point of local minima and local minimum value is <img decoding=\"async\" id=\"Picture 4907\" style=\"height: 29px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image265.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(vi)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4906\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4904\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4903\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image267.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4902\" style=\"height: 44px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image268.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4901\" style=\"height: 44px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image269.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 4900\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image270.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4899\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4898\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4897\" style=\"height: 44px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image269.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2239\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2238\" style=\"height: 27px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image271.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2237\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2236\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image164.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 2235\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/><\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" id=\"Picture 2234\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image273.png\" \/>\u00a0therefore <img decoding=\"async\" id=\"Picture 2233\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0is only the turning point.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2232\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 2231\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0is a point of local minima and local minimum value is <img decoding=\"async\" id=\"Picture 2230\" style=\"height: 41px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image274.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(vii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 2229\" style=\"height: 41px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2228\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 2227\" style=\"height: 53px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image276.png\" \/>\u00a0\u00a0and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2226\" style=\"height: 67px; width: 256px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image277.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 2225\" style=\"height: 60px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image278.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 2224\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2223\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2222\" style=\"height: 53px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image279.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2221\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2220\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0[Turning point]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 2219\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image280.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2218\" style=\"height: 60px; width: 284px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image281.png\" \/>\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2217\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 2216\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0is a point of local maxima and local maximum value is <img decoding=\"async\" id=\"Picture 2176\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image282.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(viii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4799\" style=\"height: 28px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image283.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4797\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4796\" style=\"height: 41px; width: 261px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image284.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4795\" style=\"height: 44px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image285.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4794\" style=\"height: 47px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image286.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 4793\" style=\"height: 44px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image287.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 4792\" style=\"height: 63px; width: 303px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image288.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4791\" style=\"height: 57px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image289.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4790\" style=\"height: 55px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image290.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4789\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4788\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4787\" style=\"height: 44px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image287.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4786\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4785\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image291.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4784\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4783\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image292.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4782\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4781\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image292.png\" \/>\u00a0is a point of local maxima and local maximum value is <img decoding=\"async\" id=\"Picture 4780\" style=\"height: 48px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image293.png\" \/><\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" id=\"Picture 4779\" style=\"height: 97px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image294.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4778\" style=\"height: 75px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image295.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4777\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4776\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0has local maximum value at <img decoding=\"async\" id=\"Picture 4775\" style=\"height: 41px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image296.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-3\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Prove_that_the_following_functions_do_not_have_maxima_or_minima\"><\/span><strong>4. Prove that the following functions do not have maxima or minima:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" id=\"Picture 46\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image297.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" id=\"Picture 47\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image298.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" id=\"Picture 48\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image299.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given: <img decoding=\"async\" id=\"Picture 4774\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image297.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4773\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4772\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image300.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4771\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4770\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4769\" style=\"height: 21px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\">But this gives no real value of <img decoding=\"async\" id=\"Picture 4768\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image302.png\" \/>\u00a0Therefore, there is no turning point.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4991\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4990\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0does not have maxima or minima.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4989\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image298.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4988\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4987\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image303.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4986\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4985\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4984\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image304.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4983\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a01 = 0<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"But_this_is_not_possible_Therefore_there_is_no_turning_point\"><\/span>But this is not possible. Therefore, there is no turning point.<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4982\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4981\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0does not have maxima or minima.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4980\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image299.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4979\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4978\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image305.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4977\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4976\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4975\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image306.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4974\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4973\" style=\"height: 45px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image307.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4972\" style=\"height: 45px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image308.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4971\" style=\"height: 45px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image309.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, values of <img decoding=\"async\" id=\"Picture 4970\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0are imaginary.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4969\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 4968\" style=\"height: 27px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image176.png\" \/>\u00a0does not have maxima or minima.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Find_the_absolute_maximum_value_and_the_absolute_minimum_value_of_the_following_functions_in_the_given_intervals\"><\/span><strong>5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" id=\"Picture 50\" style=\"height: 27px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image310.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" id=\"Picture 51\" style=\"height: 27px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image311.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" id=\"Picture 52\" style=\"height: 45px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image312.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <img decoding=\"async\" id=\"Picture 53\" style=\"height: 29px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image313.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4967\" style=\"height: 27px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image310.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4966\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4965\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image314.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4964\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4963\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4962\" style=\"height: 21px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image315.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4960\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4959\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image316.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4958\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image280.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4957\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image317.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4956\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image318.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4955\" style=\"height: 29px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image319.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4954\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image320.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4953\" style=\"height: 29px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image321.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, absolute minimum value of <img decoding=\"async\" id=\"Picture 4952\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 4951\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image322.png\" \/>\u00a0and absolute maximum value is 8.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 4950\" style=\"height: 27px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image311.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4948\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4947\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image323.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4946\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4945\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4944\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4943\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4942\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image213.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4941\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4939\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image215.png\" \/>\u00a0\u00a0 [Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4938\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4937\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0is in I quadrant. [<img decoding=\"async\" id=\"Picture 4936\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image324.png\" \/>\u00a0]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4935\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4934\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image325.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4933\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4932\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image218.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4931\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4930\" style=\"height: 45px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image326.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4929\" style=\"height: 27px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image327.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4928\" style=\"height: 27px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image328.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, absolute minimum value is <img decoding=\"async\" id=\"Picture 31\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image145.png\" \/>\u00a0and absolute maximum value is 1.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 30\" style=\"height: 45px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image329.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 29\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 28\" style=\"height: 41px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image330.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 27\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 26\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 25\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image331.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 24\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 23\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image332.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 22\" style=\"height: 45px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image333.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 21\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image334.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 20\" style=\"height: 41px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image335.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 19\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image318.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 18\" style=\"height: 41px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image336.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 17\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image337.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 16\" style=\"height: 49px; width: 251px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image338.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, absolute minimum value is <img decoding=\"async\" id=\"Picture 15\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image339.png\" \/>\u00a0and absolute maximum value is 8.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 14\" style=\"height: 29px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image340.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 13\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 12\" style=\"height: 27px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image341.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 11\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 10\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 9\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image342.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 8\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 7\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image343.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 6\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image260.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5\" style=\"height: 29px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image344.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image345.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2\" style=\"height: 29px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image346.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, absolute minimum value is <img decoding=\"async\" id=\"Picture 1\" style=\"height: 19px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image170.png\" \/>\u00a0and absolute maximum value is 19.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Find_the_maximum_profit_that_a_company_can_make_if_the_profit_function_is_given_by\"><\/span><strong>6. \u00a0Find the maximum profit that a company can make, if the profit function is given by <img decoding=\"async\" id=\"Picture 59\" style=\"height: 27px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image347.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Profit function <img decoding=\"async\" id=\"Picture 5299\" style=\"height: 27px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image348.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5298\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5297\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image349.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5296\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image350.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5295\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image351.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5294\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5293\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image352.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5292\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5291\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image353.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5290\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image292.png\" \/>, <img decoding=\"async\" id=\"Picture 5289\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image350.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5288\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5287\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image354.png\" \/>\u00a0has a local maximum value at <img decoding=\"async\" id=\"Picture 5286\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image292.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5285\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0At <img decoding=\"async\" id=\"Picture 5284\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image292.png\" \/>, Maximum profit<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5283\" style=\"height: 45px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image355.png\" \/><\/p>\n<p style=\"text-align: justify;\">= 41 + 16 \u2013 8 = 49<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>7. Find both the maximum value and minimum value of <img decoding=\"async\" id=\"Picture 62\" style=\"height: 21px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image356.png\" \/>\u00a0on the interval [0, 3].<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5282\" style=\"height: 27px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image357.png\" \/>\u00a0on [0, 3]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5281\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5280\" style=\"height: 27px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image358.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5279\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5278\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5277\" style=\"height: 21px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image359.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5276\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5275\" style=\"height: 21px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image360.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5274\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5273\" style=\"height: 29px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image361.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5272\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5271\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5270\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image362.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 5269\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image362.png\" \/>\u00a0is imaginary, therefore it is rejected.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5268\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5267\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0is turning point.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5266\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0At <img decoding=\"async\" id=\"Picture 5265\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image320.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5264\" style=\"height: 27px; width: 265px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image363.png\" \/>= <img decoding=\"async\" id=\"Picture 5263\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image364.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 At <img decoding=\"async\" id=\"Picture 5262\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5261\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image365.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 At <img decoding=\"async\" id=\"Picture 5260\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image263.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5259\" style=\"height: 27px; width: 301px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image366.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, absolute minimum value is <img decoding=\"async\" id=\"Picture 5258\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image364.png\" \/>\u00a0and absolute maximum value is 25.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>8. At what points on the interval <img decoding=\"async\" id=\"Picture 65\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image367.png\" \/>\u00a0does the function <img decoding=\"async\" id=\"Picture 66\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image368.png\" \/>\u00a0attain its maximum value?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5257\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image369.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5256\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5255\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image370.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5254\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5253\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5252\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image371.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5251\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5250\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image372.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5249\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5248\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image373.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" id=\"Picture 5247\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image374.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5246\" style=\"height: 41px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image375.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5245\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image369.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5244\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5243\" style=\"height: 45px; width: 195px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image376.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5242\" style=\"height: 45px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image377.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5241\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image378.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5240\" style=\"height: 29px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image379.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" id=\"Picture 5239\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image374.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5238\" style=\"height: 45px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image380.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5237\" style=\"height: 45px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image381.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5236\" style=\"height: 45px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image382.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5235\" style=\"height: 45px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image383.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also <img decoding=\"async\" id=\"Picture 5234\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image384.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5233\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image385.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" id=\"Picture 5232\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0attains its maximum value 1 at <img decoding=\"async\" id=\"Picture 5231\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image218.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5230\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image386.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the required points are <img decoding=\"async\" id=\"Picture 5229\" style=\"height: 45px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image387.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5228\" style=\"height: 45px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image388.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_What_is_the_maximum_value_of_the_function\"><\/span><strong>9. What is the maximum value of the function <img decoding=\"async\" id=\"Picture 69\" style=\"height: 19px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image389.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5227\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image390.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5226\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5225\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image323.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5224\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5223\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5222\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5221\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5220\" style=\"height: 19px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image213.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5218\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image215.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5217\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image391.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5216\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5215\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image392.png\" \/>\u00a0\u00a0[Turning point]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5214\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5213\" style=\"height: 45px; width: 276px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image393.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5212\" style=\"height: 41px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image394.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5211\" style=\"height: 44px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image395.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5210\" style=\"height: 44px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image396.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5209\" style=\"height: 29px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image397.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" id=\"Picture 5208\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image398.png\" \/>\u00a0is even, then <img decoding=\"async\" id=\"Picture 5207\" style=\"height: 45px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image399.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" id=\"Picture 5206\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image398.png\" \/>\u00a0is odd, then <img decoding=\"async\" id=\"Picture 5205\" style=\"height: 45px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image400.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, maximum value of <img decoding=\"async\" id=\"Picture 5204\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 5203\" style=\"height: 23px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image250.png\" \/>\u00a0and minimum value of <img decoding=\"async\" id=\"Picture 5202\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 5201\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image401.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Find_the_maximum_value_of_in_the_interval_1_3_Find_the_maximum_value_of_the_same_function_in\"><\/span><strong>10. Find the maximum value of <img decoding=\"async\" id=\"Picture 72\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image402.png\" \/>\u00a0in the interval [1, 3]. Find the maximum value of the same function in <img decoding=\"async\" id=\"Picture 73\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image403.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5200\" style=\"height: 27px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image404.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5199\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5198\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image405.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5197\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5196\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5195\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image406.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5194\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5193\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image407.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5192\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5191\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image408.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5190\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5189\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5188\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image164.png\" \/>\u00a0\u00a0[Turning points]\n<p style=\"text-align: justify;\">For Interval [1, 3], <img decoding=\"async\" id=\"Picture 5187\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image272.png\" \/>\u00a0is turning point.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5186\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image260.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5185\" style=\"height: 27px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image409.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5184\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image320.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5183\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image410.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5182\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image263.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5181\" style=\"height: 27px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image411.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, maximum value of <img decoding=\"async\" id=\"Picture 5180\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is 89.<\/p>\n<p style=\"text-align: justify;\">For Interval <img decoding=\"async\" id=\"Picture 5179\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image412.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5178\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image164.png\" \/>\u00a0is turning point.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5177\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image413.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5176\" style=\"height: 27px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image414.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5175\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image318.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5174\" style=\"height: 27px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image415.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5173\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image345.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5172\" style=\"height: 27px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image416.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, maximum value of <img decoding=\"async\" id=\"Picture 5171\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0is 139.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_It_is_given_that_at_the_function_attains_its_maximum_value_on_the_interval_0_20_Find_the_value_of\"><\/span><strong>11. It is given that at <img decoding=\"async\" id=\"Picture 76\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image260.png\" \/>\u00a0the function <img decoding=\"async\" id=\"Picture 77\" style=\"height: 21px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image417.png\" \/>\u00a0attains its maximum value, on the interval [0, 20]. Find the value of <img decoding=\"async\" id=\"Picture 78\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image418.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5448\" style=\"height: 27px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image419.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5447\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5446\" style=\"height: 27px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image420.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since, <img decoding=\"async\" id=\"Picture 5445\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0attains its maximum value at <img decoding=\"async\" id=\"Picture 5444\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0in the interval [0, 2], therefore <img decoding=\"async\" id=\"Picture 5443\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image421.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5442\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5441\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image422.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5440\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5439\" style=\"height: 19px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image423.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5438\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5437\" style=\"height: 19px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image424.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-4\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_Find_the_maximum_and_minimum_value_of_on\"><\/span><strong>12. Find the maximum and minimum value of <img decoding=\"async\" id=\"Picture 81\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image425.png\" \/>\u00a0on <img decoding=\"async\" id=\"Picture 82\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image426.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5436\" style=\"height: 27px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image427.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5435\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5434\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image428.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5433\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5432\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5431\" style=\"height: 19px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image429.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5430\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5429\" style=\"height: 19px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image430.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5428\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5427\" style=\"height: 41px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image431.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5426\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image432.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5425\" style=\"height: 45px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image433.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5424\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image434.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5423\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5422\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image435.png\" \/>\u00a0where <img decoding=\"async\" id=\"Picture 5421\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image436.png\" \/>\u00a0Z\u00a0 <img decoding=\"async\" id=\"Picture 5420\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5419\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image437.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5418\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image438.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5417\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image439.png\" \/>\u00a0But <img decoding=\"async\" id=\"Picture 5416\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image440.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5415\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image441.png\" \/>, therefore <img decoding=\"async\" id=\"Picture 5414\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image442.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5413\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image443.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5412\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image444.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5411\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image445.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5410\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image446.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5409\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image447.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5408\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image448.png\" \/><\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" id=\"Picture 5407\" style=\"height: 41px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image449.png\" \/>, therefore <img decoding=\"async\" id=\"Picture 5406\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image450.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, it is clear that the only turning point of <img decoding=\"async\" id=\"Picture 5405\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image130.png\" \/>\u00a0given by <img decoding=\"async\" id=\"Picture 5404\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image451.png\" \/>\u00a0which belong to given closed interval <img decoding=\"async\" id=\"Picture 5403\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image367.png\" \/>\u00a0are, <img decoding=\"async\" id=\"Picture 5402\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image452.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5401\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image442.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5400\" style=\"height: 48px; width: 323px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image453.png\" \/>\u00a0nearly<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5399\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image454.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5398\" style=\"height: 48px; width: 345px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image455.png\" \/>\u00a0nearly<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5397\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image456.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5396\" style=\"height: 48px; width: 368px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image457.png\" \/>\u00a0nearly<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5395\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image458.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5394\" style=\"height: 48px; width: 372px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image459.png\" \/>\u00a0nearly<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5393\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5392\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image460.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5391\" style=\"height: 19px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image461.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5390\" style=\"height: 27px; width: 336px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image462.png\" \/>\u00a0nearly<\/p>\n<p style=\"text-align: justify;\">Therefore, Maximum value = <img decoding=\"async\" id=\"Picture 5389\" style=\"height: 19px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image463.png\" \/>\u00a0and minimum value = 0<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-5\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Find_two_numbers_whose_sum_is_24_and_whose_product_is_as_large_as_possible\"><\/span><strong>13. Find two numbers whose sum is 24 and whose product is as large as possible.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let the two numbers be <img decoding=\"async\" id=\"Picture 5388\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5387\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image464.png\" \/><\/p>\n<p style=\"text-align: justify;\">According to the question, <img decoding=\"async\" id=\"Picture 5386\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image465.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5385\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5384\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image466.png\" \/>\u00a0\u00a0\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">And let <img decoding=\"async\" id=\"Picture 5383\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is the product of <img decoding=\"async\" id=\"Picture 5382\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5381\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image464.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5380\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5379\" style=\"height: 17px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image468.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5378\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5377\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image469.png\" \/>\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5376\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5375\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image470.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5374\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5373\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image471.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5372\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image472.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now to find turning point, <img decoding=\"async\" id=\"Picture 5371\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image473.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5370\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5369\" style=\"height: 19px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image474.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5368\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5367\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image475.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5366\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image475.png\" \/>, <img decoding=\"async\" id=\"Picture 5365\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image472.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5364\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5363\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image475.png\" \/>\u00a0is a point of local maxima and <img decoding=\"async\" id=\"Picture 5362\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is maximum at <img decoding=\"async\" id=\"Picture 5361\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image475.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5360\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i),\u00a0 <img decoding=\"async\" id=\"Picture 5359\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image476.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the two required numbers are 12 and 12.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-6\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_Find_two_positive_integers_and_such_that_and_is_maximum\"><\/span><strong>14. Find two positive integers <img decoding=\"async\" id=\"Picture 87\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 88\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0such that <img decoding=\"async\" id=\"Picture 89\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image478.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 90\" style=\"height: 24px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image479.png\" \/>\u00a0is maximum.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5358\" style=\"height: 21px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image480.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let \u00a0P = <img decoding=\"async\" id=\"Picture 5357\" style=\"height: 24px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image479.png\" \/>\u00a0\u00a0[To be maximized]\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Putting from eq. (i), <img decoding=\"async\" id=\"Picture 5356\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image481.png\" \/>\u00a0in eq. (ii),<\/p>\n<p style=\"text-align: justify;\">P = <img decoding=\"async\" id=\"Picture 5355\" style=\"height: 27px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image482.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5354\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5353\" style=\"height: 44px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image483.png\" \/>\u00a0\u2026..(iii)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5352\" style=\"height: 44px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image484.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5351\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5350\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image485.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5349\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5348\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image486.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is clear that <img decoding=\"async\" id=\"Picture 5347\" style=\"height: 44px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image487.png\" \/>\u00a0changes sign from positive to negative as <img decoding=\"async\" id=\"Picture 5346\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0increases through 45.<\/p>\n<p style=\"text-align: justify;\">Therefore, P is maximum when <img decoding=\"async\" id=\"Picture 5345\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image488.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" id=\"Picture 5344\" style=\"height: 24px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image479.png\" \/>\u00a0is maximum when <img decoding=\"async\" id=\"Picture 5343\" style=\"height: 19px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image489.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5342\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image488.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-7\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"15_Find_two_positive_integers_and_such_that_their_sum_is_35_and_the_product_is_a_maximum\"><\/span><strong>15. Find two positive integers <img decoding=\"async\" id=\"Picture 92\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 93\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0such that their sum is 35 and the product <img decoding=\"async\" id=\"Picture 94\" style=\"height: 24px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image490.png\" \/>\u00a0is a maximum.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5341\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image491.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5340\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5339\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image492.png\" \/>\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5338\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image493.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5337\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5336\" style=\"height: 29px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image494.png\" \/>\u00a0\u00a0 [From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5335\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5334\" style=\"height: 41px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image495.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5333\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5332\" style=\"height: 41px; width: 221px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image496.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5331\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5330\" style=\"height: 41px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image497.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5329\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5328\" style=\"height: 41px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image498.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5327\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5326\" style=\"height: 41px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image499.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5325\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image473.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5324\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5323\" style=\"height: 29px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image500.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5322\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5321\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5320\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image501.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5319\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image502.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5318\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5317\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5316\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image503.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5315\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image504.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5314\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image195.png\" \/>\u00a0is rejected because according to question, <img decoding=\"async\" id=\"Picture 5313\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0is a positive number.<\/p>\n<p style=\"text-align: justify;\">Also <img decoding=\"async\" id=\"Picture 5312\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image503.png\" \/>\u00a0is rejected because from eq. (i), <img decoding=\"async\" id=\"Picture 5311\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0= 35 \u2013 35 = 0, but <img decoding=\"async\" id=\"Picture 5310\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0is positive.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5309\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image504.png\" \/>\u00a0is only the turning point.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5308\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5307\" style=\"height: 44px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image505.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5306\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image504.png\" \/>, <img decoding=\"async\" id=\"Picture 5305\" style=\"height: 44px; width: 276px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image506.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5304\" style=\"height: 29px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image507.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5303\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0By second derivative test, <img decoding=\"async\" id=\"Picture 5302\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image508.png\" \/>\u00a0will be maximum at <img decoding=\"async\" id=\"Picture 5301\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image504.png\" \/>\u00a0when <img decoding=\"async\" id=\"Picture 5300\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image509.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">Therefore, the required numbers are 10 and 25.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-8\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"16_Find_two_positive_integers_whose_sum_is_16_and_sum_of_whose_cubes_is_minimum\"><\/span><strong>16. Find two positive integers whose sum is 16 and sum of whose cubes is minimum.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let the two positive numbers are <img decoding=\"async\" id=\"Picture 5608\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5607\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image464.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5606\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5605\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image510.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5604\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5603\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image511.png\" \/>\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5602\" style=\"height: 24px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image512.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5601\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5600\" style=\"height: 29px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image513.png\" \/>\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5599\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5598\" style=\"height: 29px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image514.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5597\" style=\"height: 29px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image515.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5596\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5595\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image516.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5594\" style=\"height: 44px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image517.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5593\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image508.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5592\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5591\" style=\"height: 19px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image518.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5590\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5589\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image519.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5588\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image519.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5587\" style=\"height: 44px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image517.png\" \/>\u00a0is positive.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5586\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5585\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image519.png\" \/>\u00a0is a point of local minima and <img decoding=\"async\" id=\"Picture 5584\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is minimum when <img decoding=\"async\" id=\"Picture 5583\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image519.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5582\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5581\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image520.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the required numbers are 8 and 8.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-9\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"17_A_square_piece_of_tin_of_side_18_cm_is_to_be_made_into_a_box_without_top_by_cutting_a_square_from_each_corner_and_folding_up_the_flaps_to_form_the_box_What_should_be_the_side_of_the_square_to_be_cut_off_so_that_the_volume_of_the_box_is_the_maximum_possible\"><\/span><strong>17. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Each side of square piece of tin is 18 cm.<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5580\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0cm be the side of each of the four squares cut off from each corner.<\/p>\n<p style=\"text-align: justify;\">Then dimensions of the open box formed by folding the flaps after cutting off squares are <img decoding=\"async\" id=\"Picture 5579\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image521.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5578\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0cm.<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5577\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0denotes the volume of the open box.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5576\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5575\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image522.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5574\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5573\" style=\"height: 29px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image523.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5572\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5571\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image524.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5570\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image525.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5569\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5568\" style=\"height: 41px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image526.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5567\" style=\"height: 44px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image527.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5566\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image508.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5565\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5564\" style=\"height: 21px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image528.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5562\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image529.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5561\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5560\" style=\"height: 27px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image530.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5559\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5558\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image531.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5557\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5556\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image531.png\" \/>\u00a0is rejected because at <img decoding=\"async\" id=\"Picture 5555\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image531.png\" \/>\u00a0length = <img decoding=\"async\" id=\"Picture 5554\" style=\"height: 19px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image532.png\" \/>\u00a0which is impossible.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5553\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5552\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/>\u00a0is the turning point.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5551\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/>,\u00a0 <img decoding=\"async\" id=\"Picture 5550\" style=\"height: 44px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image533.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\">\u00b7 <img decoding=\"async\" id=\"Picture 5549\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is minimum at <img decoding=\"async\" id=\"Picture 5548\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image259.png\" \/>\u00a0i.e., side of each square to be cut off from each corner for maximum volume is 3 cm.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-10\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"18_A_rectangular_sheet_of_tin_45_cm_by_24_cm_is_to_be_made_into_a_box_without_top_by_cutting_off_square_from_each_corner_and_folding_up_the_flaps_What_should_be_the_side_of_the_square_to_be_cut_off_so_that_the_volume_of_the_box_is_maximum\"><\/span><strong>18. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Dimensions of rectangular sheet are 45 cm and 24 cm.<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5547\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0cm be the side of each of the four squares cut off from each corner.<\/p>\n<p style=\"text-align: justify;\">Then dimensions of the open box formed by folding the flaps after cutting off squares are <img decoding=\"async\" id=\"Picture 5546\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image534.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5545\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0cm.<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5544\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0denotes the volume of the open box.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5543\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5542\" style=\"height: 27px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image535.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5541\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5540\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image536.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5539\" style=\"height: 21px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image537.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5538\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5537\" style=\"height: 41px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image538.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5536\" style=\"height: 44px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image539.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5535\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image508.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5534\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5533\" style=\"height: 21px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image540.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5532\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5531\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image541.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5530\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5529\" style=\"height: 27px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image542.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5528\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5527\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image543.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5526\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image544.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5525\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image544.png\" \/>\u00a0is rejected because at <img decoding=\"async\" id=\"Picture 5524\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image544.png\" \/>\u00a0length = <img decoding=\"async\" id=\"Picture 5523\" style=\"height: 19px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image545.png\" \/>\u00a0which is impossible.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5522\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5521\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image543.png\" \/>\u00a0is the turning point.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5520\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image543.png\" \/>,\u00a0 <img decoding=\"async\" id=\"Picture 5519\" style=\"height: 44px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image546.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\">\u00b7 <img decoding=\"async\" id=\"Picture 5518\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is minimum at <img decoding=\"async\" id=\"Picture 5517\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image543.png\" \/>\u00a0i.e., side of each square to be cut off from each corner for maximum volume is 5 cm.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-11\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"19_Show_that_of_all_the_rectangles_inscribed_in_a_given_fixed_circle_the_square_has_maximum_area\"><\/span><strong>19. Show that of all the rectangles inscribed in a given fixed circle, the square has maximum area.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let PQRS be the rectangle inscribed in a given circle with centre O and radius <img decoding=\"async\" id=\"Picture 5516\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image418.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 136px; width: 138px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image547.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 5515\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5514\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the length and breadth of the rectangle, i.e., <img decoding=\"async\" id=\"Picture 5513\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image548.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5512\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image549.png\" \/><\/p>\n<p style=\"text-align: justify;\">In right angled triangle PQR, using Pythagoras theorem,<\/p>\n<p style=\"text-align: justify;\">PQ<sup>2<\/sup> + QR<sup>2<\/sup> = PR<sup>2<\/sup><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5511\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5510\" style=\"height: 29px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image550.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5509\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5508\" style=\"height: 24px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image551.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5507\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5506\" style=\"height: 28px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image552.png\" \/>\u00a0\u00a0\u2026..(i)<\/p>\n<p style=\"text-align: justify;\">Let A be the area of the rectangle, then A = <img decoding=\"async\" id=\"Picture 5505\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image553.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5504\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image554.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5503\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5502\" style=\"height: 48px; width: 409px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image555.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5501\" style=\"height: 48px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image556.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 5500\" style=\"height: 75px; width: 323px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image557.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5499\" style=\"height: 65px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image558.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5498\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5497\" style=\"height: 61px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image559.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5496\" style=\"height: 65px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image560.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5495\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image561.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5494\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5493\" style=\"height: 48px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image556.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5492\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5491\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image562.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5490\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/><img decoding=\"async\" id=\"Picture 5489\" style=\"height: 23px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image563.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5488\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0At <img decoding=\"async\" id=\"Picture 5487\" style=\"height: 23px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image563.png\" \/>, <img decoding=\"async\" id=\"Picture 5486\" style=\"height: 55px; width: 289px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image564.png\" \/>\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5485\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0At <img decoding=\"async\" id=\"Picture 5484\" style=\"height: 23px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image563.png\" \/>, area of rectangle is maximum.<\/p>\n<p style=\"text-align: justify;\">And from eq. (i), <img decoding=\"async\" id=\"Picture 5483\" style=\"height: 28px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image565.png\" \/>,<\/p>\n<p style=\"text-align: justify;\">i.e., <img decoding=\"async\" id=\"Picture 5482\" style=\"height: 25px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image566.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the area of inscribed rectangle is maximum when it is square.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-12\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"20_Show_that_the_right_circular_cylinder_of_given_surface_and_maximum_volume_is_such_that_its_height_is_equal_to_the_diameter_of_the_base\"><\/span><strong>20. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5481\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0be the radius of the circular base and <img decoding=\"async\" id=\"Picture 5480\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the height of closed right circular cylinder.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5479\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Total surface area (S) = <img decoding=\"async\" id=\"Picture 5478\" style=\"height: 24px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image567.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5477\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5476\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image568.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5475\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image569.png\" \/>\u00a0(say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5474\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5473\" style=\"height: 24px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image570.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5472\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5471\" style=\"height: 44px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image571.png\" \/>\u00a0\u00a0\u2026..(i)<\/p>\n<p style=\"text-align: justify;\">Volume of cylinder <img decoding=\"async\" id=\"Picture 5470\" style=\"height: 27px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image572.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5469\" style=\"height: 51px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image573.png\" \/>\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5468\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5467\" style=\"height: 29px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image574.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5466\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5465\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image575.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5464\" style=\"height: 44px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image576.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5463\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image473.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5462\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5461\" style=\"height: 29px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image577.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5460\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5459\" style=\"height: 47px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image578.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5458\" style=\"height: 47px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image578.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5457\" style=\"height: 47px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image579.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5456\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5455\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is maximum at <img decoding=\"async\" id=\"Picture 5454\" style=\"height: 47px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image578.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5453\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 5452\" style=\"height: 84px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image580.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5451\" style=\"height: 47px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image581.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5450\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image582.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5449\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0Height = Diameter<\/p>\n<p style=\"text-align: justify;\">Therefore, the volume of cylinder is maximum when its height is equal to the diameter of its base.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-13\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"21_Of_all_the_closed_cylindrical_cans_right_circular_of_a_given_volume_of_100_cubic_centimeters_find_the_dimensions_of_the_can_which_has_the_minimum_surface_area\"><\/span><strong>21. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimeters, find the dimensions of the can which has the minimum surface area.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5756\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0be the radius of the circular base and <img decoding=\"async\" id=\"Picture 5755\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the height of closed right circular cylinder.<\/p>\n<p style=\"text-align: justify;\">According to the question, Volume of the cylinder <img decoding=\"async\" id=\"Picture 5754\" style=\"height: 24px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image583.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5753\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5752\" style=\"height: 41px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image584.png\" \/>\u00a0\u2026(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5751\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Total surface area (S) = <img decoding=\"async\" id=\"Picture 5750\" style=\"height: 24px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image567.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5749\" style=\"height: 29px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image585.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5748\" style=\"height: 45px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image586.png\" \/>\u00a0\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5747\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0S = <img decoding=\"async\" id=\"Picture 5746\" style=\"height: 45px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image587.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5745\" style=\"height: 45px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image588.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5744\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5743\" style=\"height: 45px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image589.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5742\" style=\"height: 47px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image590.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5741\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image591.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5740\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5739\" style=\"height: 45px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image592.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5738\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5737\" style=\"height: 45px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image593.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5736\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5735\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image594.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5734\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5733\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image595.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5732\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5731\" style=\"height: 55px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image596.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5730\" style=\"height: 55px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image596.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5729\" style=\"height: 91px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image597.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5728\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image598.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5727\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image599.png\" \/>\u00a0\u00a0\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5726\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0S is minimum when radius <img decoding=\"async\" id=\"Picture 5725\" style=\"height: 55px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image596.png\" \/>\u00a0cm<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5724\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i)<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" id=\"Picture 5723\" style=\"height: 75px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image600.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5722\" style=\"height: 55px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image601.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5721\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image582.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-14\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"22_A_wire_of_length_28_m_is_to_be_cut_into_two_pieces_One_of_the_pieces_is_to_be_made_into_a_square_and_the_other_into_a_circle_What_should_be_the_length_of_the_two_pieces_so_that_the_combined_area_of_the_square_and_the_circle_is_minimum\"><\/span><strong>22. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5720\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0meters be the side of square and <img decoding=\"async\" id=\"Picture 5719\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0meters be the radius of the circle.<\/p>\n<p style=\"text-align: justify;\">Length of the wire = Perimeter of square + Circumference of circle<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5718\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5717\" style=\"height: 21px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image602.png\" \/>\u00a0= 28<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5716\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5715\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image603.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5714\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5713\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image604.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Area of square = <img decoding=\"async\" id=\"Picture 5712\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image605.png\" \/>\u00a0and Area of circle = <img decoding=\"async\" id=\"Picture 5711\" style=\"height: 24px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image606.png\" \/><\/p>\n<p style=\"text-align: justify;\">Combined area (A) = <img decoding=\"async\" id=\"Picture 5710\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image605.png\" \/>\u00a0+ <img decoding=\"async\" id=\"Picture 5709\" style=\"height: 24px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image606.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5708\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image605.png\" \/>\u00a0+ <img decoding=\"async\" id=\"Picture 5707\" style=\"height: 49px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image607.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5706\" style=\"height: 41px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image608.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5705\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5704\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image609.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5703\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image610.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5702\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image561.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5701\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5700\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image611.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5699\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5698\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image612.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5697\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5696\" style=\"height: 19px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image613.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5695\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5694\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image614.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5693\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5692\" style=\"height: 41px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image615.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 5691\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image610.png\" \/>\u00a0\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5690\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0A is minimum when <img decoding=\"async\" id=\"Picture 5689\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image616.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the wire should be cut at a distance <img decoding=\"async\" id=\"Picture 5688\" style=\"height: 41px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image617.png\" \/>\u00a0from one end.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-15\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"23_Prove_that_the_volume_of_the_largest_cone_that_can_be_inscribed_in_a_sphere_of_radius_R_is_of_the_volume_of_the_sphere\"><\/span><strong>23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is <img decoding=\"async\" id=\"Picture 129\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image618.png\" \/>\u00a0of the volume of the sphere.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let O be the centre and R be the radius of the given sphere, BM = <img decoding=\"async\" id=\"Picture 5687\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0and AM = <img decoding=\"async\" id=\"Picture 5686\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 124px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image619.png\" \/><\/p>\n<p style=\"text-align: justify;\">In right angled triangle OMB, using Pythagoras theorem,<\/p>\n<p style=\"text-align: justify;\">OM<sup>2<\/sup> + BM<sup>2<\/sup> = OB<sup>2<\/sup><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5685\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5684\" style=\"height: 29px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image620.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5683\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5682\" style=\"height: 24px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image621.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5681\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5680\" style=\"height: 24px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image622.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5679\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5678\" style=\"height: 24px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image623.png\" \/>\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Volume of a cone inscribed in the given sphere<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" id=\"Picture 5677\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image624.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5676\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image625.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5675\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5674\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image626.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5673\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5672\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image627.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5671\" style=\"height: 44px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image628.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5670\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image473.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5669\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5668\" style=\"height: 41px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image629.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5667\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5666\" style=\"height: 24px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image630.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5665\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5664\" style=\"height: 24px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image631.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5663\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5662\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image632.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5661\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image632.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5660\" style=\"height: 47px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image633.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5659\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image634.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5658\" style=\"height: 41px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image635.png\" \/>\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5657\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5656\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is maximum at <img decoding=\"async\" id=\"Picture 5655\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image632.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5654\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i) <img decoding=\"async\" id=\"Picture 5653\" style=\"height: 49px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image636.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5652\" style=\"height: 44px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image637.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5651\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Maximum volume of the cone<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5650\" style=\"height: 44px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image638.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5649\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image618.png\" \/>\u00a0(Volume of the sphere)<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-16\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"24_Show_that_the_right_circular_cone_of_least_curve_surface_and_given_volume_has_an_altitude_equal_to_time_the_radius_of_the_base\"><\/span><strong>24. Show that the right circular cone of least curve surface and given volume has an altitude equal to <img decoding=\"async\" id=\"Picture 136\" style=\"height: 23px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image250.png\" \/>\u00a0time the radius of the base.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5648\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0be the radius and <img decoding=\"async\" id=\"Picture 5647\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the height of the cone.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5646\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Volume of the cone (V) = <img decoding=\"async\" id=\"Picture 5645\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image639.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5644\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5643\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image640.png\" \/>\u00a0(say) \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">And Surface area of the cone (S) = <img decoding=\"async\" id=\"Picture 5642\" style=\"height: 29px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image641.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5641\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5640\" style=\"height: 29px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image642.png\" \/>\u00a0(say) \u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5639\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5638\" style=\"height: 48px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image643.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5637\" style=\"height: 48px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image644.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5636\" style=\"height: 29px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image645.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5635\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5634\" style=\"height: 44px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image646.png\" \/>\u00a0\u00a0and <img decoding=\"async\" id=\"Picture 5633\" style=\"height: 47px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image647.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5632\" style=\"height: 44px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image648.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5631\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5630\" style=\"height: 29px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image649.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5629\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5628\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image650.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5627\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5626\" style=\"height: 44px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image651.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5625\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/><img decoding=\"async\" id=\"Picture 5624\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image652.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5623\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5622\" style=\"height: 35px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image653.png\" \/>\u00a0\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5621\" style=\"height: 35px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image653.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5620\" style=\"height: 57px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image654.png\" \/>\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5619\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5618\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is minimum when <img decoding=\"async\" id=\"Picture 5617\" style=\"height: 35px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image653.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5616\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 5615\" style=\"height: 55px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image655.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5614\" style=\"height: 67px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image656.png\" \/>\u00a0[From eq. (iii)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5613\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5612\" style=\"height: 24px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image657.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5611\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5610\" style=\"height: 25px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image658.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, Surface area is minimum when height = <img decoding=\"async\" id=\"Picture 5609\" style=\"height: 23px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image250.png\" \/>\u00a0(radius of base)<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-17\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"25_Show_that_the_semi-vertical_angle_of_the_cone_of_the_maximum_value_and_of_given_slant_height_is\"><\/span><strong>25. Show that the semi-vertical angle of the cone of the maximum value and of given slant height is <img decoding=\"async\" id=\"Picture 144\" style=\"height: 23px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image659.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5843\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0be the radius, <img decoding=\"async\" id=\"Picture 5842\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the height, <img decoding=\"async\" id=\"Picture 5841\" style=\"height: 19px; width: 9px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image660.png\" \/>\u00a0be the slant height of given cone and <img decoding=\"async\" id=\"Picture 5840\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image661.png\" \/>\u00a0be the semi-vertical angle of cone.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5839\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5838\" style=\"height: 24px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image662.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5837\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5836\" style=\"height: 24px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image663.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5835\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Volume of the cone (V) = <img decoding=\"async\" id=\"Picture 5834\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image639.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5833\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0V = <img decoding=\"async\" id=\"Picture 5832\" style=\"height: 41px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image664.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5831\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image665.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5830\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5829\" style=\"height: 44px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image666.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5828\" style=\"height: 47px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image667.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5827\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image668.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5826\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5825\" style=\"height: 41px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image669.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5824\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5823\" style=\"height: 24px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image670.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5822\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5821\" style=\"height: 24px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image671.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5820\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5819\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image672.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5818\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image672.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5817\" style=\"height: 48px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image673.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5816\" style=\"height: 44px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image674.png\" \/>\u00a0\u00a0\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5815\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0V is maximum at <img decoding=\"async\" id=\"Picture 5814\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image672.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5813\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 5812\" style=\"height: 44px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image675.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5811\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5810\" style=\"height: 44px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image676.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5809\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Semi-vertical angle, <img decoding=\"async\" id=\"Picture 5808\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image677.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5807\" style=\"height: 85px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image678.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5806\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5805\" style=\"height: 23px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image679.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-18\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"26_Show_that_the_semi-vertical_angle_of_the_right_circular_cone_of_given_surface_area_and_maximum_volume_is\"><\/span><strong>26. Show that the semi-vertical angle of the right circular cone of given surface area and maximum volume is <img decoding=\"async\" id=\"Picture 151\" style=\"height: 45px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image680.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5804\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image216.png\" \/>\u00a0be the radius and <img decoding=\"async\" id=\"Picture 5803\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image477.png\" \/>\u00a0be the height of the cone and semi-vertical angle be <img decoding=\"async\" id=\"Picture 5802\" style=\"height: 19px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image681.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5801\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Total Surface area of cone (S) = <img decoding=\"async\" id=\"Picture 5800\" style=\"height: 29px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image682.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5799\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5798\" style=\"height: 41px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image683.png\" \/>\u00a0(say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5797\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5796\" style=\"height: 29px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image684.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5795\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5794\" style=\"height: 32px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image685.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5793\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5792\" style=\"height: 24px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image686.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5791\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5790\" style=\"height: 24px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image687.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5789\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5788\" style=\"height: 47px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image688.png\" \/>\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Volume of cone (V) = <img decoding=\"async\" id=\"Picture 5787\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image639.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5786\" style=\"height: 51px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image689.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5785\" style=\"height: 48px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image690.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5784\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5783\" style=\"height: 44px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image691.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5782\" style=\"height: 67px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image692.png\" \/>\u00a0\u00a0[Using quotient rule]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5781\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5780\" style=\"height: 60px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image693.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5779\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image668.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5778\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5777\" style=\"height: 60px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image694.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5776\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5775\" style=\"height: 24px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image695.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5774\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5773\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image696.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5772\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5771\" style=\"height: 25px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image697.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5770\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5769\" style=\"height: 25px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image698.png\" \/>\u00a0\u00a0[height can\u2019t be negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5768\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5767\" style=\"height: 25px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image698.png\" \/>\u00a0is the turning point.<\/p>\n<p style=\"text-align: justify;\">Since, <img decoding=\"async\" id=\"Picture 5766\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image699.png\" \/>, therefore, Volume is maximum at <img decoding=\"async\" id=\"Picture 5765\" style=\"height: 25px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image698.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5764\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 5763\" style=\"height: 44px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image700.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5762\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 5761\" style=\"height: 45px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image701.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now Semi-vertical angle of the cone <img decoding=\"async\" id=\"Picture 5760\" style=\"height: 49px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image702.png\" \/>= <img decoding=\"async\" id=\"Picture 5759\" style=\"height: 87px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image703.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5758\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5757\" style=\"height: 41px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image704.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>Choose the correct answer in the Exercises 27 to 29.<\/strong><\/p>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-19\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"27_The_point_on_the_curve_which_is_nearest_to_the_point_0_5_is\"><\/span><strong>27. The point on the curve <img decoding=\"async\" id=\"Picture 160\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image705.png\" \/>\u00a0which is nearest to the point (0, 5) is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 161\" style=\"height: 32px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image706.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 162\" style=\"height: 32px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image707.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) (0, 0)\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) (2, 2)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curve is <img decoding=\"async\" id=\"Picture 5907\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image705.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let P<img decoding=\"async\" id=\"Picture 5906\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image708.png\" \/>\u00a0be any point on the curve (i), then according to question,<\/p>\n<p style=\"text-align: justify;\">Distance between given point (0, 5) and P = <img decoding=\"async\" id=\"Picture 5905\" style=\"height: 35px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image709.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5903\" style=\"height: 13px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image710.png\" \/>\u00a0(say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5902\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5901\" style=\"height: 29px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image711.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5900\" style=\"height: 29px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image712.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5899\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5898\" style=\"height: 24px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image713.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5897\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5896\" style=\"height: 24px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image714.png\" \/>\u00a0= Z (say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5895\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5894\" style=\"height: 44px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image715.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5893\" style=\"height: 47px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image716.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5892\" style=\"height: 44px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image717.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5891\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5890\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image718.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5889\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5888\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image719.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5855\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image719.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5854\" style=\"height: 47px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image716.png\" \/>\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5853\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0Z is minimum and <img decoding=\"async\" id=\"Picture 5852\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image467.png\" \/>\u00a0is minimum at <img decoding=\"async\" id=\"Picture 5851\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image719.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5850\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0From eq. (i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5849\" style=\"height: 21px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image720.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5848\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5847\" style=\"height: 23px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image721.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5846\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5845\" style=\"height: 32px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image706.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5844\" style=\"height: 32px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image722.png\" \/>\u00a0are two points on curve (i) which are nearest to (0, 5).<\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>28. For all real values of <img decoding=\"async\" id=\"Picture 168\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image723.png\" \/>\u00a0the minimum value of <img decoding=\"async\" id=\"Picture 169\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image724.png\" \/>\u00a0is:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(A) 0\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) 1\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) 3\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 170\" style=\"height: 41px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image725.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5928\" style=\"height: 44px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image726.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5927\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5926\" style=\"height: 72px; width: 395px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image727.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5925\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5924\" style=\"height: 60px; width: 324px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image728.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5923\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5922\" style=\"height: 56px; width: 409px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image729.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5921\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5920\" style=\"height: 56px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image730.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5219\" style=\"height: 60px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image731.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4961\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4949\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4940\" style=\"height: 60px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image731.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4917\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4908\" style=\"height: 29px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image732.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4905\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5919\" style=\"height: 21px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image733.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5918\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5917\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image734.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5916\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5915\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image735.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5914\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5913\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5912\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/>\u00a0[Turning points]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5911\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image172.png\" \/>,<\/p>\n<p style=\"text-align: justify;\">from eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5910\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image736.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5909\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>,<\/p>\n<p style=\"text-align: justify;\">from eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5908\" style=\"height: 41px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image737.png\" \/>\u00a0[Minimum value]\n<p style=\"text-align: justify;\">Therefore, option (D) is correct.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>29. The maximum value of <img decoding=\"async\" id=\"Picture 174\" style=\"height: 35px; width: 162px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image738.png\" \/>\u00a0is:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 175\" style=\"height: 51px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image739.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 176\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image740.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) 1\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 177\" style=\"height: 41px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image725.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 178\" style=\"height: 37px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image741.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 179\" style=\"height: 40px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image742.png\" \/>, <img decoding=\"async\" id=\"Picture 180\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image743.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 181\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 182\" style=\"height: 44px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image744.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 183\" style=\"height: 61px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image745.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 184\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 185\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 186\" style=\"height: 61px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image745.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 187\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 188\" style=\"height: 19px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image131.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 189\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image128.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 190\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image132.png\" \/>\u00a0\u00a0 [Turning point] and it belongs to the given enclosed interval <img decoding=\"async\" id=\"Picture 191\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image743.png\" \/>\u00a0i.e., [0, 1].<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 192\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image132.png\" \/>, from eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 193\" style=\"height: 55px; width: 300px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image746.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 194\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image280.png\" \/>\u00a0\u00a0from eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 195\" style=\"height: 35px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image747.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 196\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image149.png\" \/>,\u00a0 from eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 197\" style=\"height: 35px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image748.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 198\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image191.png\" \/>\u00a0\u00a0Maximum value of <img decoding=\"async\" id=\"Picture 199\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.5\/image749.png\" \/>\u00a0is 1.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_65-20\"><\/span>NCERT Solutions class 12 Maths Exercise 6.5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Class_12\"><\/span>CBSE App for Class 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n<div id=\"wp_rp_first\" class=\"wp_rp_wrap wp_rp_plain\">\n<div class=\"wp_rp_content\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 6.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 6.5\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 6.5\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4548","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide\" \/>\n<meta property=\"og:description\" content=\"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\" \/>\n<meta property=\"og:site_name\" content=\"myCBSEguide\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/mycbseguide\/\" \/>\n<meta property=\"article:published_time\" content=\"2016-05-16T06:19:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2018-05-04T09:29:12+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/09\/mycbseguide_n.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"599\" \/>\n\t<meta property=\"og:image:height\" content=\"242\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"myCBSEguide\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@mycbseguide\" \/>\n<meta name=\"twitter:site\" content=\"@mycbseguide\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"myCBSEguide\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"13 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\"},\"author\":{\"name\":\"myCBSEguide\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/#\/schema\/person\/10b8c7820ff29025ab8524da7c025f65\"},\"headline\":\"NCERT Solutions class 12 Maths Exercise 6.5\",\"datePublished\":\"2016-05-16T06:19:00+00:00\",\"dateModified\":\"2018-05-04T09:29:12+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\"},\"wordCount\":2675,\"commentCount\":5,\"publisher\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/#organization\"},\"image\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\",\"keywords\":[\"CBSE Study Material\",\"class 12\",\"Mathematics\",\"NCERT Solutions\"],\"articleSection\":[\"Mathematics\",\"NCERT Solutions\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\",\"url\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\",\"name\":\"NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide\",\"isPartOf\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\",\"datePublished\":\"2016-05-16T06:19:00+00:00\",\"dateModified\":\"2018-05-04T09:29:12+00:00\",\"description\":\"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.\",\"breadcrumb\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage\",\"url\":\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\",\"contentUrl\":\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mycbseguide.com\/blog\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"NCERT Solutions\",\"item\":\"https:\/\/mycbseguide.com\/blog\/category\/ncert-solutions\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"NCERT Solutions class 12 Maths Exercise 6.5\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/#website\",\"url\":\"https:\/\/mycbseguide.com\/blog\/\",\"name\":\"myCBSEguide\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mycbseguide.com\/blog\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/#organization\",\"name\":\"myCBSEguide\",\"url\":\"https:\/\/mycbseguide.com\/blog\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/04\/books_square.png\",\"contentUrl\":\"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/04\/books_square.png\",\"width\":180,\"height\":180,\"caption\":\"myCBSEguide\"},\"image\":{\"@id\":\"https:\/\/mycbseguide.com\/blog\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/mycbseguide\/\",\"https:\/\/x.com\/mycbseguide\",\"https:\/\/www.linkedin.com\/company\/mycbseguide\/\",\"http:\/\/in.pinterest.com\/mycbseguide\/\",\"https:\/\/www.youtube.com\/channel\/UCxuqSnnygFzwJG0pwogCNEQ\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mycbseguide.com\/blog\/#\/schema\/person\/10b8c7820ff29025ab8524da7c025f65\",\"name\":\"myCBSEguide\",\"sameAs\":[\"http:\/\/mycbseguide.com\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide","description":"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/","og_locale":"en_US","og_type":"article","og_title":"NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide","og_description":"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.","og_url":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/","og_site_name":"myCBSEguide","article_publisher":"https:\/\/www.facebook.com\/mycbseguide\/","article_published_time":"2016-05-16T06:19:00+00:00","article_modified_time":"2018-05-04T09:29:12+00:00","og_image":[{"width":599,"height":242,"url":"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/09\/mycbseguide_n.jpg","type":"image\/jpeg"}],"author":"myCBSEguide","twitter_card":"summary_large_image","twitter_creator":"@mycbseguide","twitter_site":"@mycbseguide","twitter_misc":{"Written by":"myCBSEguide","Est. reading time":"13 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#article","isPartOf":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/"},"author":{"name":"myCBSEguide","@id":"https:\/\/mycbseguide.com\/blog\/#\/schema\/person\/10b8c7820ff29025ab8524da7c025f65"},"headline":"NCERT Solutions class 12 Maths Exercise 6.5","datePublished":"2016-05-16T06:19:00+00:00","dateModified":"2018-05-04T09:29:12+00:00","mainEntityOfPage":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/"},"wordCount":2675,"commentCount":5,"publisher":{"@id":"https:\/\/mycbseguide.com\/blog\/#organization"},"image":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage"},"thumbnailUrl":"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg","keywords":["CBSE Study Material","class 12","Mathematics","NCERT Solutions"],"articleSection":["Mathematics","NCERT Solutions"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/","url":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/","name":"NCERT Solutions class 12 Maths Exercise 6.5 | myCBSEguide","isPartOf":{"@id":"https:\/\/mycbseguide.com\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage"},"image":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage"},"thumbnailUrl":"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg","datePublished":"2016-05-16T06:19:00+00:00","dateModified":"2018-05-04T09:29:12+00:00","description":"NCERT Solutions class 12 Maths Exercise 6.5 in PDF format for free download. NCERT Solutions Class 12 Maths.","breadcrumb":{"@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#primaryimage","url":"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg","contentUrl":"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg"},{"@type":"BreadcrumbList","@id":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-5\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mycbseguide.com\/blog\/"},{"@type":"ListItem","position":2,"name":"NCERT Solutions","item":"https:\/\/mycbseguide.com\/blog\/category\/ncert-solutions\/"},{"@type":"ListItem","position":3,"name":"NCERT Solutions class 12 Maths Exercise 6.5"}]},{"@type":"WebSite","@id":"https:\/\/mycbseguide.com\/blog\/#website","url":"https:\/\/mycbseguide.com\/blog\/","name":"myCBSEguide","description":"","publisher":{"@id":"https:\/\/mycbseguide.com\/blog\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mycbseguide.com\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mycbseguide.com\/blog\/#organization","name":"myCBSEguide","url":"https:\/\/mycbseguide.com\/blog\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mycbseguide.com\/blog\/#\/schema\/logo\/image\/","url":"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/04\/books_square.png","contentUrl":"https:\/\/mycbseguide.com\/blog\/wp-content\/uploads\/2016\/04\/books_square.png","width":180,"height":180,"caption":"myCBSEguide"},"image":{"@id":"https:\/\/mycbseguide.com\/blog\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/mycbseguide\/","https:\/\/x.com\/mycbseguide","https:\/\/www.linkedin.com\/company\/mycbseguide\/","http:\/\/in.pinterest.com\/mycbseguide\/","https:\/\/www.youtube.com\/channel\/UCxuqSnnygFzwJG0pwogCNEQ"]},{"@type":"Person","@id":"https:\/\/mycbseguide.com\/blog\/#\/schema\/person\/10b8c7820ff29025ab8524da7c025f65","name":"myCBSEguide","sameAs":["http:\/\/mycbseguide.com"]}]}},"_links":{"self":[{"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/posts\/4548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/comments?post=4548"}],"version-history":[{"count":5,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/posts\/4548\/revisions"}],"predecessor-version":[{"id":14879,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/posts\/4548\/revisions\/14879"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/"}],"wp:attachment":[{"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/media?parent=4548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/categories?post=4548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mycbseguide.com\/blog\/wp-json\/wp\/v2\/tags?post=4548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}