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# NCERT Solutions class 12 Maths Exercise 6.2

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## 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.

### 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.

### 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 .

Therefore,  is neither increasing nor decreasing in

### 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

### 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

### 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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### 3 thoughts on “NCERT Solutions class 12 Maths Exercise 6.2”

1. 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.

2. Ans14. -4

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