Question 21
Which of the following functions is decreasing on (0,π/2)
(A) sin 2x (B) tan x
(C) cos x (D) cos 3x
To check decreasing, we check if π^β² (π)<π in (0,π/2)
Option A
π(π₯)=π ππ 2π₯
Differentiating w.r.t. π
π^β² (π)=2 πππ 2π₯
Let 2π = ΞΈ
β΄ fβ(π₯) = 2 cos ΞΈ
When
0 < x < π/2 , then 0 < ΞΈ < π
Now,
For 0 < ΞΈ < π /π
cos ΞΈ > 0
Putting π=2π₯
cosβ‘2π₯>0
2 cosβ‘2π₯>0
β΄ π^β² (π)>π
For π /π < ΞΈ < π
cos ΞΈ < 0
Putting π=2π₯
cosβ‘2π₯<0
2 cosβ‘2π₯<0
β΄ π^β² (π)<π
So, sinβ‘2π₯ is neither increasing nor decreasing in the interval (0,π/2).
Option B
π(π₯)=π‘ππ π₯
Differentiating w.r.t. π
fβ(π) = sec2 π₯
As square of any number is always positive
So, fβ(π₯) > 0 for all values of π₯
β΄ f is strictly increasing on (0 , π/2).
Option C
π(π₯)=πππ π₯
Differentiating w.r.t. π
π^β² (π)=βπ ππ π₯
Since,
sin π > 0 for π₯ β (0 , π/2)
So,
β sin π < 0 for π₯ β (0 , π/2)
β΄ fβ (π₯) < 0 for π₯ β (0 , π/2)
So, f is strictly decreasing in (0 , π/2).
Option D
π(π₯)=πππ 3π₯
Differentiating w.r.t. π
fβ (π) = β3 sin 3π₯
Let 3π = ΞΈ
β΄ fβ (π₯) = β3 sin ΞΈ
When
0 < x < π/2 , then 0 < ΞΈ < ππ /π
For 0 < ΞΈ < π
sin ΞΈ > 0
Putting π=3π₯
sinβ‘3π₯>0
β3 sinβ‘3π₯<0
β΄ π^β² (π)<π
For π < ΞΈ < ππ /π
sin ΞΈ < 0
Putting π=3π₯
sinβ‘3π₯<0
β3 sinβ‘3π₯>0
β΄ π^β² (π)>π
So, cos 3π₯ is neither increasing nor decreasing in the interval (0,π/2).
Hence, only πππ π is decreasing in the interval (0,π/2).
So, the correct answer is (C).

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.