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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Application of Derivatives

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Application of Derivatives

Let f(x)=4 ^3 x+3 3 ^2 x-10 . The number of points of local maxima...

Let f(x)=4cos⁡3x+33cos⁡2x−10f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10. The number of points of local maxima of ff in interval (0,2π)(0,2 \pi) isLet f(x)=4 ^3 x+3 3 ^2 x-10 . The number of points of local maxima of f in interval (0,2 π) is

Options

  1. A3
  2. B4
  3. C1
  4. D2

Correct answer

D. 2

Step-by-step solution

f(x)=4cos⁡3(x)+33cos⁡2(x)−10;x∈(0,2π)⇒f′(x)=12cos⁡2x[−sin⁡(x)]+33(2cos⁡(x))[−sin⁡(x)]⇒f′(x)=−6sin⁡(x)cos⁡(x)[2cos⁡(x)+3]\begin{aligned} & \mathrm{f}(\mathrm{x})=4 \cos ^3(\mathrm{x})+3 \sqrt{3} \cos ^2(\mathrm{x})-10 ; \mathrm{x} \in(0,2 \pi) \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=12 \cos ^2 \mathrm{x}[-\sin (\mathrm{x})]+3 \sqrt{3}(2 \cos (\mathrm{x}))[-\sin (\mathrm{x})] \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-6 \sin (\mathrm{x}) \cos (\mathrm{x})[2 \cos (\mathrm{x})+\sqrt{3}]\end{aligned} Let f(x)=4 ^3 x+3 3 ^2 x-10 . The number of points of local maxima of f in interval (0,2 π) is
local maxima at x=5π6,7π6\mathrm{x}=\frac{5 \pi}{6}, \frac{7 \pi}{6}
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