NCERT Solutions class 12 Maths Exercise 6.2

NCERT Solutions class 12 Maths Exercise 6.2 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 Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Applications of Derivatives as PDF.

NCERT Solutions for Class 12 Maths Application of Derivatives 

1. Show that the function given by  is strictly increasing on R.

Ans. Given:

   i.e., positive for all  R

Therefore,  is strictly increasing on R.

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NCERT Solutions class 12 Maths Exercise 6.2

2. Show that the function given by  is strictly increasing on R.

Ans. Given:

  =  > 0  i.e., positive for all  R

Therefore,  is strictly increasing on R.

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NCERT Solutions class 12 Maths Exercise 6.2

3. Show that the function given by  is (a) strictly increasing  (b) strictly decreasing in  (c) neither increasing nor decreasing in  

Ans. Given:

 

(a) Since,  > 0, i.e., positive in first quadrant, i.e., in

Therefore,  is strictly increasing in

(b) Since,  < 0, i.e., negative in second quadrant, i.e., in

Therefore,  is strictly decreasing in

(c) Since  > 0, i.e., positive in first quadrant, i.e., in  and  < 0, i.e., negative in second quadrant, i.e., in  and .

   does not have the same sign in the interval

Therefore,  is neither increasing nor decreasing in

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NCERT Solutions class 12 Maths Exercise 6.2

4. Find the intervals in which the function  given by  is (a) strictly increasing, (b) strictly decreasing.

Ans. Given:

      ……….(i)

Now

 

Therefore, we have two intervals  and

(a) For interval  taking  (say), then from eq. (i),  > 0.

Therefore,  is strictly increasing in

(b) For interval  taking  (say), then from eq. (i),  < 0.

Therefore,  is strictly decreasing in

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NCERT Solutions class 12 Maths Exercise 6.2

5. Find the intervals in which the function  given by  is (a) strictly increasing, (b) strictly decreasing.

Ans. (a) Given:

   =

    ……….(i)

Now

  or

  or

Therefore, we have sub-intervals are  and

For interval    taking  (say), from eq. (i),

 > 0

Therefore,  is strictly increasing in

For interval    taking  (say), from eq. (i),

 < 0

Therefore,  is strictly decreasing in

For interval    taking  (say), from eq. (i),

 > 0

Therefore,  is strictly increasing in

Hence, (a)  is strictly increasing in  and

(b)  is strictly decreasing in

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NCERT Solutions class 12 Maths Exercise 6.2

6. Find the intervals in which the following functions are strictly increasing or decreasing:

(a)    

(b)    

(c)  

(d)    

(e)  

Ans. (a) Given:

     ……….(i)

Now

 

Therefore, we have two sub-intervals  and

For interval  taking  (say), from eq. (i),  < 0

Therefore,  is strictly decreasing.

For interval  taking  (say), from eq. (i),  > 0

Therefore,  is strictly increasing.

(b) Given:

  =   ……….(i)

Now

 

Therefore, we have two sub-intervals  and

For interval  taking  (say), from eq. (i),

 > 0

Therefore,  is strictly increasing.

For interval  taking  (say), from eq. (i),

 < 0

Therefore,  is strictly decreasing.

(c) Given:

 

 

=  ……….(i)

Now  = 0

  or

Therefore, we have three disjoint intervals  and

For interval , from eq. (i),

 =  < 0

Therefore,  is strictly decreasing.

For interval , from eq. (i),

 =  > 0

Therefore,  is strictly increasing.

For interval , from eq. (i),

 =  < 0

Therefore,  is strictly decreasing.

(d) Given:

  

Now

 

Therefore, we have three disjoint intervals  and

For interval , 

Therefore,  is strictly increasing.

For interval , 

Therefore,  is strictly decreasing.

(e) Given:

 

 

 

 

Here, factors  and  are non-negative for all

Therefore,  is strictly increasing if

 

 

And  is strictly decreasing if

 

 

Hence,  is strictly increasing in  and  is strictly decreasing in

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7. Show that  is an increasing function of  throughout its domain.

Ans. Given:

 

   =

=

=

 

=  ……….(i)

Domain of the given function is given to be

 

Also  and

  From eq. (i),  for all  in domain  and  is an increasing function.

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8. Find the value of  for which  is an increasing function.

Ans. Given:  

 

 

[Applying Product Rule]

 

=

=    ……….(i)

 

Therefore, we have

For   taking  (say),

 

  is decreasing.

For   taking  (say),

 

  is increasing.

For   taking  (say),

 

  is decreasing.

For   taking  (say),

 

  is increasing.

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9. Prove that  is an increasing function of  in  

Ans. Given:

 

=

 

=

 

=

=

Since  and we have , therefore

   for

Hence,  is an increasing function of  in

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10. Prove that the logarithmic function is strictly increasing on  

Ans. Given:

  for all  in

Therefore,  is strictly increasing on

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11. Prove that the function  given by  is neither strictly increasing nor strictly decreasing on  

Ans. Given:

 

 is strictly increasing if

 

 

i.e., increasing on the interval

 is strictly decreasing if

 

 

i.e., decreasing on the interval

hence,  is neither strictly increasing nor decreasing on the interval

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12. Which of the following functions are strictly decreasing on  

Ans. (A)

 

Since  in  therefore

 

Therefore,  is strictly decreasing on

(B)

 

Since

    therefore

 

Therefore,  is strictly decreasing on

(C)

 

Since

 

For   

 

Therefore,  is strictly decreasing on

For   

 

Therefore,  is strictly increasing on

Hence,  is neither strictly increasing not strictly decreasing on

(D)

  > 0

Therefore,  is strictly increasing on

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13. On which of the following intervals is the function  given by  is strictly decreasing:

(A) (0, 1) 

(B)    

(C)    

(D) None of these

Ans. Given:

  

(A) On (0, 1),    therefore

And for

  (0, 1 radian) =  > 0

Therefore,  is strictly increasing on (0, 1).

(B) For   

= = (1.5, 3.1) > 1 and hence  > 100

For   is in second quadrant and hence  is negative and between  and 0.

Therefore,  is strictly increasing on .

(C) On  = (0, 1.5) both terms of given function are positive.

Therefore,  is strictly increasing on .

(D) Option (D) is the correct answer.

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14. Find the least value of  such that the function  given by  strictly increasing on (1, 2).

Ans.

  

Since  is strictly increasing on (1, 2), therefore  > 0 for all  in (1, 2)

 On (1, 2)

 

 

 Minimum value of  is  and maximum value is

Since  > 0 for all  in (1, 2)

  and

  and

Therefore least value of  is

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15. Let I be any interval disjoint from  Prove that the function  given by  is strictly increasing on I.

Ans. Given:

  

    ……….(i)

Here for every  either  or

  for ,  (say),

 > 0

And for ,  (say),

 > 0

  > 0 for all  , hence  is strictly increasing on I.

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16. Prove that the function  given by  is strictly increasing on  and strictly decreasing on  

Ans. Given:

 

On the interval  i.e., in first quadrant,

 > 0

Therefore,  is strictly increasing on .

On the interval  i.e., in second quadrant,

 < 0

Therefore,  is strictly decreasing on .

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17. Prove that the function  given by  is strictly decreasing on  and strictly decreasing on

Ans. Given:

 

On the interval  i.e., in first quadrant,  is positive, thus  < 0

Therefore,  is strictly decreasing on .

On the interval  i.e., in second quadrant,  is negative thus  > 0

Therefore,  is strictly increasing on .

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18. Prove that the function given by  is increasing in R.

Ans. Given:

 

  for all  in R.

Therefore,  is increasing on R.

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19. The interval in which  is increasing in:

(A)    

(B)    

(C)    

(D) (0, 2)

Ans. Given:

 

=

 

=

 

In option (D),  for all  in the interval (0, 2).

Therefore, option (D) is correct.

NCERT Solutions class 12 Maths Exercise 6.2

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

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