Question

Prove that the function $f$ given by $f\left(x\right)=\log \sin x$ is strictly increasing on $(0,\frac{\pi }{2})$ and strictly decreasing on $(\frac{\pi }{2},\pi )$

Answer 1

We have, $f\left(x\right)=\log \sin x$
$\therefore f^{\prime }\left(x\right)=\frac{1}{\sin x}\cos x=cotx$
In interval $(0,\frac{\pi }{2}),f^{\prime }\left(x\right)=\cot x>0$.
$\therefore$ $f$ is strictly increasing in $(0,\frac{\pi }{2})$.
In interval $(\frac{\pi }{2},\pi ),f^{\prime }\left(x\right)=\cot x<0.$
$\therefore$ $f$ is strictly decreasing in $(\frac{\pi }{2},\pi )$.