Question 8
Which of the following functions is decreasing on (0,đ/2)
(A) sin 2x (B) tan x
(C) cos x (D) cos 3x
To check decreasing, we check if đ^â˛ (đ)<đ in (0,đ/2)
Option A
đ(đĽ)=đ đđ 2đĽ
Differentiating w.r.t. đ
đ^â˛ (đ)=2 đđđ 2đĽ
Let 2đ = Î¸
â´ fâ(đĽ) = 2 cos Î¸
When
0 < x < đ/2 , then 0 < Î¸ < đ
Now,
So, sinâĄ2đĽ is neither increasing nor decreasing in the interval (0,đ/2).
Option B
đ(đĽ)=đĄđđ đĽ
Differentiating w.r.t. đ
fâ(đ) = sec2 đĽ
For 0 < Î¸ < đ /đ
cos Î¸ > 0
Putting đ=2đĽ
cosâĄ2đĽ>0
2 cosâĄ2đĽ>0
â´ đ^â˛ (đ)>đ
For đ /đ < Î¸ < đ
cos Î¸ < 0
Putting đ=2đĽ
cosâĄ2đĽ<0
2 cosâĄ2đĽ<0
â´ đ^â˛ (đ)<đ
As square of any number is always positive
So, fâ(đĽ) > 0 for all values of đĽ
â´ f is strictly increasing on (0 , đ/2).
Option C
đ(đĽ)=đđđ đĽ
Differentiating w.r.t. đ
đ^â˛ (đ)=âđ đđ đĽ
Since,
sin đ > 0 for đĽ â (0 , đ/2)
So,
â sin đ < 0 for đĽ â (0 , đ/2)
â´ fâ (đĽ) < 0 for đĽ â (0 , đ/2)
So, f is strictly decreasing in (0 , đ/2).
Option D
đ(đĽ)=đđđ 3đĽ
Differentiating w.r.t. đ
fâ (đ) = â3 sin 3đĽ
Let 3đ = Î¸
â´ fâ (đĽ) = â3 sin Î¸
When
0 < x < đ/2 , then 0 < Î¸ < đđ /đ
For 0 < Î¸ < đ
sin Î¸ > 0
Putting đ=3đĽ
sinâĄ3đĽ>0
â3 sinâĄ3đĽ<0
â´ đ^â˛ (đ)<đ
For đ < Î¸ < đđ /đ
sin Î¸ < 0
Putting đ=3đĽ
sinâĄ3đĽ<0
â3 sinâĄ3đĽ>0
â´ đ^â˛ (đ)>đ
So, cos 3đĽ is neither increasing nor decreasing in the interval (0,đ/2).
Hence, only đđđ đ is decreasing in the interval (0,đ/2).
So, the correct answer is (C).

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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