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/2) (A) sin 2x (B) tan x (C) cos x (D) cos 3x To check decreasing, we check if 𝒇^′ (𝒙)<𝟎 in (0,𝜋/2) Option A 𝑓(𝑥)=𝑠𝑖𝑛 2𝑥 Differentiating w.r.t. 𝒙 𝒇^′ (𝒙)=2 𝑐𝑜𝑠 2𝑥 Let 2𝒙 = θ ∴ f’(𝑥) = 2 cos θ When 0 < x < 𝜋/2 , then 0 < θ < 𝜋 Now, So, sin⁡2𝑥 is neither increasing nor decreasing in the interval (0,𝜋/2). Option B 𝑓(𝑥)=𝑡𝑎𝑛 𝑥 Differentiating w.r.t. 𝒙 f’(𝒙) = sec2 𝑥 For 0 < θ < 𝝅/𝟐 cos θ > 0 Putting 𝜃=2𝑥 cos⁡2𝑥>0 2 cos⁡2𝑥>0 ∴ 𝒇^′ (𝒙)>𝟎 For 𝝅/𝟐 < θ < 𝝅 cos θ < 0 Putting 𝜃=2𝑥 cos⁡2𝑥<0 2 cos⁡2𝑥<0 ∴ 𝒇^′ (𝒙)<𝟎 As square of any number is always positive So, f’(𝑥) > 0 for all values of 𝑥 ∴ f is strictly increasing on (0 , 𝜋/2). Option C 𝑓(𝑥)=𝑐𝑜𝑠 𝑥 Differentiating w.r.t. 𝒙 𝒇^′ (𝒙)=−𝑠𝑖𝑛 𝑥 Since, sin 𝒙 > 0 for 𝑥 ∈ (0 , 𝜋/2) So, – sin 𝒙 < 0 for 𝑥 ∈ (0 , 𝜋/2) ∴ f’ (𝑥) < 0 for 𝑥 ∈ (0 , 𝜋/2) So, f is strictly decreasing in (0 , 𝜋/2). Option D 𝑓(𝑥)=𝑐𝑜𝑠 3𝑥 Differentiating w.r.t. 𝒙 f’ (𝒙) = –3 sin 3𝑥 Let 3𝒙 = θ ∴ f’ (𝑥) = –3 sin θ When 0 < x < 𝜋/2 , then 0 < θ < 𝟑𝝅/𝟐 For 0 < θ < 𝝅 sin θ > 0 Putting 𝜃=3𝑥 sin⁡3𝑥>0 −3 sin⁡3𝑥<0 ∴ 𝒇^′ (𝒙)<𝟎 For 𝝅 < θ < 𝟑𝝅/𝟐 sin θ < 0 Putting 𝜃=3𝑥 sin⁡3𝑥<0 −3 sin⁡3𝑥>0 ∴ 𝒇^′ (𝒙)>𝟎 So, cos 3𝑥 is neither increasing nor decreasing in the interval (0,𝜋/2). Hence, only 𝒄𝒐𝒔 𝒙 is decreasing in the interval (0,𝜋/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.