Chapter 6 Application of Derivatives Ex 6.2 Solutions
Question - 11 : - Which of the followingfunctions are strictly decreasing on
?(A) cos x (B) cos 2x (C)cos 3x (D) tan x
Answer - 11 : - (A) Let
In interval
is strictlydecreasing in interval
.
(B) Let
is strictlydecreasing in interval.
(C) Let
The point 
divides the intervalinto two disjoint intervals i.e., 0
.
∴ f3 isstrictly increasing in interval
Hence, f3 isneither increasing nor decreasing in interval
(D) Let
In interval
∴ f4 isstrictly increasing in interval
Therefore, functionscos x and cos 2x are strictly decreasing in Hence, the correct answers are A and B.
strictly decreasing?
(A) 
(B) 
(C) 
(D) None of these
Answer - 12 : -
We have,
In interval
Thus, function f isstrictly increasing in interval (0, 1).
In interval
Thus, function f isstrictly increasing in interval
∴ f is strictly increasing in interval
Hence, function f isstrictly decreasing in none of the intervals.
The correct answer is D.
Question - 13 : - Find the least valueof a such that the function f given 
is strictly increasing on [1, 2].
Answer - 13 : -
We have,
Now, function f isincreasing on [1,2].
Question - 14 : - Let I be any interval disjoint from (−1, 1). Prove that thefunction f given by
is strictlyincreasing on I.
Answer - 14 : -
We have,
The points x =1 and x = −1 divide the real line in three disjoint intervalsi.e., 
In interval (−1, 1), it is observed that:
∴ f is strictly decreasing on 
In intervals
, it is observed that:
∴ f is strictly increasing on
Hence, function f is strictlyincreasing in interval I disjointfrom (−1, 1).
Hence, the given result is proved.
Question - 15 : - Prove that thefunction f given by f(x) = log sin x isstrictly increasing on 
and strictly decreasing on
Answer - 15 : -
We have,
In interval
∴ f is strictly increasing in
In interval
∴f isstrictly decreasing in
Question - 16 : - Prove that thefunction f given by f(x) = log cos x isstrictly decreasing on and strictlyincreasing on
Answer - 16 : -
We have,
In interval
∴f isstrictly decreasing on
In interval
∴f isstrictly increasing on
Question - 17 : - Prove that the functiongiven by 
is increasing in R.
Answer - 17 : - 
For any x∈R, (x −1)2 > 0.
Thus, 
is always positive in R.Hence, the given function (f) isincreasing in R.
Question - 18 : - The interval in which 
is increasing is
(A)
(B) (−2, 0) (C) 
(D) (0, 2)
Answer - 18 : -
We have,
The points x =0 and x = 2 divide the real line into three disjoint intervalsi.e.,
In intervals
is always positive.
∴f isdecreasing on
In interval (0, 2),
∴ f is strictly increasing on (0, 2).
Hence, f is strictlyincreasing in interval (0, 2).
The correct answer is D.