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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2018 · Mathematics · Application of Derivatives

Which one of the following is correct in respect of the function...

Which one of the following is correct in respect of the function f(x)=xsin⁡x+cos⁡x+12cos⁡2x?f(x)=x \sin x+\cos x+\frac{1}{2} \cos ^2 x ?

Options

  1. AIt is increasing in the interval (0,π2)\left(0, \frac{\pi}{2}\right)
  2. BIt remain constant in the interval (0,π2)\left(0, \frac{\pi}{2}\right)
  3. CIt is decreasing in the interval (0,π2)\left(0, \frac{\pi}{2}\right)
  4. DIt is decreasing in the interval (π4,π2)\left(\frac{\pi}{4}, \frac{\pi}{2}\right)

Correct answer

A. It is increasing in the interval (0,π2)\left(0, \frac{\pi}{2}\right)

Step-by-step solution

f(x)=xsin⁡x+cos⁡x+12cos⁡2xf′(x)=xcos⁡x+sin⁡x−sin⁡x−sin⁡xcos⁡x=cos⁡x(x−sin⁡x)>0 in (0,π2)\begin{aligned} & f(x)=x \sin x+\cos x+\frac{1}{2} \cos ^2 x \\ & f^{\prime}(x)=x \cos x+\sin x-\sin x-\sin x \cos x \\ & =\cos x(x-\sin x) > 0 \text{ in }\left(0, \frac{\pi}{2}\right)\end{aligned}$$
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