Which of the following functions is decreasing on (0,π/2)

(A) sin 2x                           (B) tan x

(C) cos x                            (D) cos 3x

 

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  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 21 Which of the following functions is decreasing on (0,πœ‹/2) (A) sin 2x (B) tan x (C) cos x (D) cos 3x To check decreasing, we check if 𝒇^β€² (𝒙)<𝟎 in (0,πœ‹/2) Option A 𝑓(π‘₯)=𝑠𝑖𝑛 2π‘₯ Differentiating w.r.t. 𝒙 𝒇^β€² (𝒙)=2 π‘π‘œπ‘  2π‘₯ Let 2𝒙 = ΞΈ ∴ f’(π‘₯) = 2 cos ΞΈ When 0 < x < πœ‹/2 , then 0 < ΞΈ < πœ‹ Now, For 0 < ΞΈ < 𝝅/𝟐 cos ΞΈ > 0 Putting πœƒ=2π‘₯ cos⁑2π‘₯>0 2 cos⁑2π‘₯>0 ∴ 𝒇^β€² (𝒙)>𝟎 For 𝝅/𝟐 < ΞΈ < 𝝅 cos ΞΈ < 0 Putting πœƒ=2π‘₯ cos⁑2π‘₯<0 2 cos⁑2π‘₯<0 ∴ 𝒇^β€² (𝒙)<𝟎 So, sin⁑2π‘₯ is neither increasing nor decreasing in the interval (0,πœ‹/2). Option B 𝑓(π‘₯)=π‘‘π‘Žπ‘› π‘₯ Differentiating w.r.t. 𝒙 f’(𝒙) = sec2 π‘₯ As square of any number is always positive So, f’(π‘₯) > 0 for all values of π‘₯ ∴ f is strictly increasing on (0 , πœ‹/2). Option C 𝑓(π‘₯)=π‘π‘œπ‘  π‘₯ Differentiating w.r.t. 𝒙 𝒇^β€² (𝒙)=βˆ’π‘ π‘–π‘› π‘₯ Since, sin 𝒙 > 0 for π‘₯ ∈ (0 , πœ‹/2) So, – sin 𝒙 < 0 for π‘₯ ∈ (0 , πœ‹/2) ∴ f’ (π‘₯) < 0 for π‘₯ ∈ (0 , πœ‹/2) So, f is strictly decreasing in (0 , πœ‹/2). Option D 𝑓(π‘₯)=π‘π‘œπ‘  3π‘₯ Differentiating w.r.t. 𝒙 f’ (𝒙) = –3 sin 3π‘₯ Let 3𝒙 = ΞΈ ∴ f’ (π‘₯) = –3 sin ΞΈ When 0 < x < πœ‹/2 , then 0 < ΞΈ < πŸ‘π…/𝟐 For 0 < ΞΈ < 𝝅 sin ΞΈ > 0 Putting πœƒ=3π‘₯ sin⁑3π‘₯>0 βˆ’3 sin⁑3π‘₯<0 ∴ 𝒇^β€² (𝒙)<𝟎 For 𝝅 < ΞΈ < πŸ‘π…/𝟐 sin ΞΈ < 0 Putting πœƒ=3π‘₯ sin⁑3π‘₯<0 βˆ’3 sin⁑3π‘₯>0 ∴ 𝒇^β€² (𝒙)>𝟎 So, cos 3π‘₯ is neither increasing nor decreasing in the interval (0,πœ‹/2). Hence, only 𝒄𝒐𝒔 𝒙 is decreasing in the interval (0,πœ‹/2). So, the correct answer is (C).

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Davneet Singh
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.