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Install NowNCERT 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.

### 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.

### 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

### 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

### 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

### 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

**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.

**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.

**9. Prove that is an increasing function of in **

**Ans. **Given:

=

=

=

=

Since and we have , therefore

for

Hence, is an increasing function of in

**10. Prove that the logarithmic function is strictly increasing on **

**Ans. **Given:

for all in

Therefore, is strictly increasing on

**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

**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

**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.

**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

**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.

**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 .

**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 .

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

**Ans. **Given:

for all in R.

Therefore, is increasing on R.

**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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The answers solved by this site is not matching with the NCERT textbook answers. I’m confused which ans, method or procedure I should follow.

Pls sort this problem.

Ans14. -4

Solution is good but try to solve little typically to know question easily