NCERT Exemplar - MCQs

Chapter 6 Class 12 Application of Derivatives
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## (C) decreasing in ((-Ο)/2 ", "Β  Ο/2)Β  `Β  Β  Β  Β  Β  (D) decreasingΒ  in ("0, "Β  Ο/2)

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Question 20 The function f (x) = 4 sin3 x β 6 sin2 x + 12 sin x + 100 is strictly Increasing in (π, 3π/2) (B) decreasing in (π/2,π) (C) decreasing in ((βπ)/2 ", " π/2) ` (D) decreasing in ("0, " π/2) f (π₯) = 4 sin^3β‘γπ₯β6 sin^2β‘γπ₯+12 sinβ‘γπ₯+100γ γ γ Differentiating w.r.t π fβ (π) = 4 Γβ‘γ3 γβ‘γγπ ππγ^2 π₯γ Γ (sinβ‘γπ₯)β²γβ6 Γ 2 sinβ‘π₯Γ (sinβ‘π₯ )β²+12 cosβ‘π₯+0 = 12 sin^2 π₯ cos π₯ β 12 sin π₯ cos π₯ + 12 cos π₯ = 12 cos π₯ (γsin^2 π₯γβ‘γβsinβ‘γπ₯+1γ γ ) = 12 cos π (γγπππγ^π πγβ‘γ+ (πβπ¬π’π§β‘π )γ ) Now, we need to check sign of fβ (π) Checking sign of γπππγ^πβ‘γπ+(γπβπππγβ‘π ) γ Since, γπππγ^π π β₯ 0 And β1 β€ sin π β€ 1 1 β₯ β sin π₯ β₯ β1 1 + 1 β₯ 1 β sin π₯ β₯ 1 β 1 2 β₯ 1 β sin π₯ β₯ 0 0 β€ 1 β sin π₯ β€ 2 So, (1 β sin π) β₯π Therefore, γπππγ^πβ‘γπ+(γπβπππγβ‘π ) γβ₯ 0 Either cos πβ₯ 0 π₯ π(0, π/2)βͺ(3π/2, 2π) i.e., 1st and 4th quadrant, which can be written as π β ((βπ)/π,π/π) Or cos π β€ 0 π₯ π(π/2,π)βͺ(π,3π/2) i.e., 2nd and 3rd quadrant, which can be written as π β(π/π,ππ/π) Thus, f is increasing in the interval ((βπ)/π,π/π) f is decreasing in the interval (π/π,ππ/π) Our options are (A) Increasing in (π, 3π/2) (B) decreasing in (π/2,π) (C) decreasing in ((βπ)/2 ", " π/2) ` (D) decreasing in ("0, " π/2) As (π/2,π) lies in the interval (π/2,3π/2) So, f is decreasing in this interval (π/π,π) So, the correct answer is (B)