Q92 / 155 · WBJEE 2020 · Mathematics

In open interval $\left(0, \frac{\pi}{2}\right)$,

  • A
    $\cos x+x \sin x < 1$
  • Correct answer
    B
    $\cos x+x \sin x>1$
  • C
    no specific order relation can be ascertained between $\cos x+x \sin x$ and 1
  • D
    $\cos x+x \sin x < \frac{1}{2}$

Solution

Hint: $f(x)=\cos x+x \sin x-1$ $\Rightarrow f^{\prime}(x)=-\sin x+\sin x+x \cos x>0 ; x \in\left(0, \frac{\pi}{2}\right)$ $\Rightarrow f(x)$ is increasing function $\begin{array}{l} \Rightarrow f(x)>f(0) \\ \cos x+x \sin x-1>0 \\ \cos x+x \sin x>1 \end{array}$

Hence, the correct answer is option: 'B'.

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