The function f (x) = 4 sin 3 x – 6 sin 2 x + 12 sin x + 100 is strictly

(A)Increasing in (π, 3π/2)                          (B) decreasing in (π/2,π)

(C) decreasing in ((-π)/2 ", "  π/2)  `          (D) decreasing  in ("0, "  π/2)

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

Transcript

Question 20 The function f (x) = 4 sin3 x – 6 sin2 x + 12 sin x + 100 is strictly Increasing in (πœ‹, 3πœ‹/2) (B) decreasing in (πœ‹/2,πœ‹) (C) decreasing in ((βˆ’πœ‹)/2 ", " πœ‹/2) ` (D) decreasing in ("0, " πœ‹/2) f (π‘₯) = 4 sin^3⁑〖π‘₯βˆ’6 sin^2⁑〖π‘₯+12 sin⁑〖π‘₯+100γ€— γ€— γ€— Differentiating w.r.t 𝒙 f’ (𝒙) = 4 ×⁑〖3 〗⁑〖〖𝑠𝑖𝑛〗^2 π‘₯γ€— Γ— (sin⁑〖π‘₯)β€²γ€—βˆ’6 Γ— 2 sin⁑π‘₯Γ— (sin⁑π‘₯ )β€²+12 cos⁑π‘₯+0 = 12 sin^2 π‘₯ cos π‘₯ βˆ’ 12 sin π‘₯ cos π‘₯ + 12 cos π‘₯ = 12 cos π‘₯ (γ€–sin^2 π‘₯γ€—β‘γ€–βˆ’sin⁑〖π‘₯+1γ€— γ€— ) = 12 cos 𝒙 (γ€–γ€–π’”π’Šπ’γ€—^𝟐 𝒙〗⁑〖+ (πŸβˆ’π¬π’π§β‘π’™ )γ€— ) Now, we need to check sign of f’ (𝒙) Checking sign of γ€–π’”π’Šπ’γ€—^πŸβ‘γ€–π’™+(γ€–πŸβˆ’π’”π’Šπ’γ€—β‘π’™ ) γ€— Since, γ€–π’”π’Šπ’γ€—^𝟐 𝒙 β‰₯ 0 And βˆ’1 ≀ sin 𝒙 ≀ 1 1 β‰₯ βˆ’ sin π‘₯ β‰₯ βˆ’1 1 + 1 β‰₯ 1 βˆ’ sin π‘₯ β‰₯ 1 βˆ’ 1 2 β‰₯ 1 βˆ’ sin π‘₯ β‰₯ 0 0 ≀ 1 βˆ’ sin π‘₯ ≀ 2 So, (1 βˆ’ sin 𝒙) β‰₯𝟎 Therefore, γ€–π’”π’Šπ’γ€—^πŸβ‘γ€–π’™+(γ€–πŸβˆ’π’”π’Šπ’γ€—β‘π’™ ) γ€—β‰₯ 0 Either cos 𝒙β‰₯ 0 π‘₯ πœ–(0, πœ‹/2)βˆͺ(3πœ‹/2, 2πœ‹) i.e., 1st and 4th quadrant, which can be written as 𝒙 ∈ ((βˆ’π…)/𝟐,𝝅/𝟐) Or cos 𝒙 ≀ 0 π‘₯ πœ–(πœ‹/2,πœ‹)βˆͺ(πœ‹,3πœ‹/2) i.e., 2nd and 3rd quadrant, which can be written as 𝒙 ∈(𝝅/𝟐,πŸ‘π…/𝟐) Thus, f is increasing in the interval ((βˆ’π…)/𝟐,𝝅/𝟐) f is decreasing in the interval (𝝅/𝟐,πŸ‘π…/𝟐) Our options are (A) Increasing in (πœ‹, 3πœ‹/2) (B) decreasing in (πœ‹/2,πœ‹) (C) decreasing in ((βˆ’πœ‹)/2 ", " πœ‹/2) ` (D) decreasing in ("0, " πœ‹/2) As (πœ‹/2,πœ‹) lies in the interval (πœ‹/2,3πœ‹/2) So, f is decreasing in this interval (𝝅/𝟐,𝝅) So, the correct answer is (B)

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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.