# NCERT Solutions class 12 Maths Exercise 6.2 (Ex 6.2) Chapter 6 Application of Derivatives

## NCERT Solutions for Class 12 Maths Exercise 6.2 Chapter 6 Application of Derivatives – FREE PDF Download

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.2 (Ex 6.2) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 6 Application of Derivatives Exercise 6.2 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

# NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives (Ex 6.2) Exercise 6.2

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 disjoint sub  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 three disjoint sub-intervals   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:

……….(i)

Now

Therefore, we have two disjoint intervals  and

For interval ,  taking x=-6 (say)

from eq. (i),   > 0

Therefore,  is strictly increasing.

For interval ,    taking x=  0   (say)

from eq. (i),      <  0

Therefore,  is strictly decreasing.

(e) Given:

Here, factors  and  are non-negative for all x

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:

=   11+x(4+2x2x)(2+x)211+x−(4+2x−2x)(2+x)2

……….(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 four disjoint subintervals

For   taking  (say),

is decreasing.

For   taking  (say),

is increasing.

For   taking  (say),

is decreasing.

For   taking  (say),

is increasing.

thus  is increasing in  and

is decreasing in  and

### 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

thus for  x  in       sinx is positive in first quadrent

So,

Therefore,  is strictly decreasing on

(B)

Since

therefore

So,

Therefore,  is strictly decreasing on

(C)

Since

thus two cases    and

For

So,

Therefore,  is strictly decreasing on

For

So ,

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  xIx∈I, hence  is strictly increasing on  II

### 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 increasing 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.