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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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Question 8 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, 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‘₯ 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 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 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.