Question

Assertion :$\frac{2}{\pi }<\frac{\sin x}{x}<1$ for $0\le \left|x\right|\le \pi /2$ Reason: $f\left(x\right)=\frac{\sin x}{x},0<x\le \pi /2$ is decreasing.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

consider, $f\left(x\right)=\frac{\sin x}{x}$ where $0\le \left|x\right|\le \pi /2$
$f^{\prime }\left(x\right)=\frac{x\cos x-\sin x}{x^2}$
let $u\left(x\right)=x\cos x-\sin x$
$\Rightarrow u^{\prime }\left(x\right)=-x\sin x<0$ for $x\in (0,\pi /2)$
Therefore, $u\left(x\right)$ is a decreasing function
Since, $x\ge 0$ and $u\left(x\right)$ is a decreasing function
Therefore, $u\left(x\right)<u\left(0\right)=0$
thus $f^{\prime }\left(x\right)<0$
hence, $f\left(x\right)$ is a decreasing function.
and since, $0\le \left|x\right|\le \pi /2$
Therefore, $f\left(\frac{\pi }{2}\right)<f\left(x\right)<f\left(0\right)$
$\Rightarrow \frac{2}{\pi }<\frac{\sin x}{x}<1$