Question

The function $f\left(x\right)=\frac{\sin x}{x}$ is decreasing in the interval

• A. $(-\frac{\pi }{2},0)$
• B. $(0,\frac{\pi }{2})$
• C. $(-\frac{\pi }{4},0)$
• D. None of these

Answer 1

Answer: (B)

$(0,\frac{\pi }{2})$

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

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

Let $g\left(x\right)=x\cos x-\sin x$

$=\cos x(x-\tan x)$

where $\cos x>0$

$x-\tan x<0$

$\Rightarrow g\left(x\right)<0$

$\Rightarrow f^{\prime }\left(x\right)<0$

$\therefore f\left(n\right)$ is a decreasing function in $(0,\frac{\pi }{2})$