Question 8
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,
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₯
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
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).

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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