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.
i.e., positive for all 
R
Therefore, 
is strictly increasing on R.
2. Show that the function given by 
is strictly increasing on R.
= 
> 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
(c) neither increasing nor decreasing in
(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
4. Find the intervals in which the function given by is (a) strictly increasing, (b) strictly decreasing.
……….(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.
= 
……….(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) 
……….(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.
is an increasing function of
= 11+x−(4+2x−2x)(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.
is an increasing function.
[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
= 
= 
= 
= 
Since 
and we have 
, therefore 
for
Hence, 
is an increasing function of 
in 
10. Prove that the logarithmic function is strictly increasing on
for all in 
Therefore, is strictly increasing on
11. Prove that the function given by is neither strictly increasing nor strictly decreasing on
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
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
(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).
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.
Prove that the function 
is strictly increasing on I.
……….(i)
Here for every either 
or
for , (say),
> 0
And for , (say),
> 0
> 0 for all x∈Ix∈I, hence is strictly increasing on II
16. Prove that the function given by is strictly increasing on and strictly decreasing on
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
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.
for all in R.
Therefore, is increasing on R.
19. The interval in which is increasing in:
is increasing in:
(A) 
(B) 
(C)
(D) (0, 2)
= 
= 
In option (D), 
for all in the interval (0, 2).
Therefore, option (D) is correct.
