Discuss the nature of the function f given by $f\left(x\right) = \log \left[\sin \left(x\right)\right]$ in the intervals $\left(0,\dfrac{\pi}{2}\right)$ and $\left(\dfrac{\pi}{2}, \pi\right)$.
$f\left(x\right) = \log \left[\sin \left(x\right)\right]$
$\therefore$ $f'\left(x\right) = \dfrac{\cos x}{\sin x} = \cot x$
For $0 < x < \dfrac{\pi}{2}$, $\cot x > 0$
i.e. $f'\left(x\right) > 0$ for $0 < x < \dfrac{\pi}{2}$
i.e. $f\left(x\right)$ is strictly increasing in the interval $\left(0,\dfrac{\pi}{2}\right)$
For $\dfrac{\pi}{2} < x < \pi$, $\cot x < 0$
i.e. $f'\left(x\right) < 0$ for $\dfrac{\pi}{2} < x < \pi$
i.e. $f\left(x\right)$ is strictly decreasing in the interval $\left(\dfrac{\pi}{2}, \pi\right)$