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Chapter 6 Class 12 Application of Derivatives
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## (C) cos xΒ  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  (D) cos 3x

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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^β² (π)<π in (0,π/2) Option A π(π₯)=π ππ 2π₯ Differentiating w.r.t. π π^β² (π)=2 πππ  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‘ππ 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^β² (π)>π For π/π < ΞΈ < π cos ΞΈ < 0 Putting π=2π₯ cosβ‘2π₯<0 2 cosβ‘2π₯<0 β΄ π^β² (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₯)=πππ  π₯ Differentiating w.r.t. π π^β² (π)=βπ ππ 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₯)=πππ  3π₯ Differentiating w.r.t. π fβ (π) = β3 sin 3π₯ Let 3π = ΞΈ β΄ fβ (π₯) = β3 sin ΞΈ When 0 < x < π/2 , then 0 < ΞΈ < ππ/π For 0 < ΞΈ < π sin ΞΈ > 0 Putting π=3π₯ sinβ‘3π₯>0 β3 sinβ‘3π₯<0 β΄ π^β² (π)<π For π < ΞΈ < ππ/π sin ΞΈ < 0 Putting π=3π₯ sinβ‘3π₯<0 β3 sinβ‘3π₯>0 β΄ π^β² (π)>π So, cos 3π₯ is neither increasing nor decreasing in the interval (0,π/2). Hence, only πππ π is decreasing in the interval (0,π/2). So, the correct answer is (C).