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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2022 · Mathematics · Application of Derivatives

How many extreme values does 4 x+2 x , where 0 < x < (π)/(2) have?

How many extreme values does sin⁡4x+2x\sin 4 x+2 x, where 0<x<π20 < x < \frac{\pi}{2} have?

Options

  1. A1
  2. B2
  3. C4
  4. D8

Correct answer

B. 2

Step-by-step solution

 Let f(x)=sin⁡4x+2xf′(x)=4cos⁡4x+2=0\begin{aligned} & \text{ Let } \mathrm{f}(\mathrm{x})=\sin 4 \mathrm{x}+2 \mathrm{x} \\ & \mathrm{f}^{\prime}(\mathrm{x})=4 \cos 4 \mathrm{x}+2=0\end{aligned}⇒cos⁡4x=−12=cos⁡(π−π3)⇒4x=2nπ±2π3⇒x=nπ2±π6x=π6,π3[∵0<x<π2]\begin{aligned} & \Rightarrow \cos 4 \mathrm{x}=-\frac{1}{2}=\cos \left(\pi-\frac{\pi}{3}\right) \\ & \Rightarrow 4 \mathrm{x}=2 \mathrm{n} \pi \pm \frac{2 \pi}{3} \\ & \Rightarrow \mathrm{x}=\mathrm{n} \frac{\pi}{2} \pm \frac{\pi}{6} \\ & \mathrm{x}=\frac{\pi}{6}, \frac{\pi}{3}\left[\because 0 < \mathrm{x} < \frac{\pi}{2}\right]\end{aligned}
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