Question

Assertion :The function $f\left(x\right)=\frac{\sin x}{x}$ is decreasing in the interval $(0,\frac{\pi }{2})$ Reason: $\tan x>x$ for $0<x<\frac{\pi }{2}$

• 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 incorrect but Reason is correct
• 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

We have, $f\left(x\right)=\frac{\sin x}{x}$

$\Rightarrow f^{\prime }\left(x\right)=\frac{x\cos x-\sin x}{x^2}=\frac{\cos x(x-\tan x)}{x^2}$

But $\tan x>x$ and $\cos x>0,$ for $0<x<\frac{\pi }{2}$

$\therefore f^{\prime }\left(x\right)<0$ in the interval $(0,\frac{\pi }{2})$

Thus, $f\left(x\right)$ is decreasing in $(0,\frac{\pi }{2})$