Question

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

Answer 1

We have $f\left(x\right)=\log \cos x$

Differentiate both sides wrt x to get

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

comparing it to the graph of $h\left(x\right)=\tan x$ in the intervals $(0,\frac{\pi }{2})and\left(\frac{\pi }{2},\pi \right)$

we can say that the given function is strictly decreasing on $(0,\frac{\pi }{2})$ and strictly increasing on $\left(\frac{\pi }{2},\pi \right)$