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.
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
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