Answer
$ a.\quad$ critical points at $x=\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{2\pi}{3}$ , $\displaystyle \frac{4\pi}{3}$
$ b.\quad$ f is increasing on $(\displaystyle \frac{2\pi}{3}, \frac{4\pi}{3})$, decreasing on $(0, \displaystyle \frac{\pi}{2})\cup(\frac{\pi}{2}, \frac{2\pi}{3})\cup(\frac{4\pi}{3}$, $2\pi)$
$ c.\quad$ Local maximum at $x=\displaystyle \frac{4\pi}{3}$ and $x=0$,
local minimum at $x=\displaystyle \frac{2\pi}{3}$ and $ x=2\pi$
Work Step by Step
$(a)$
$f'$ is defined on $[0,2\pi]$
$f'(x)=0$ for
$\sin x=1\Rightarrow x=\pi/2,$
$\displaystyle \cos x=-\frac{1}{2}\Rightarrow x=\frac{2\pi}{3}, x=\frac{4\pi}{3}$
Critical points at $x=\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{2\pi}{3}$ , $\displaystyle \frac{4\pi}{3}$
$(b)$
Calculate $f'$ at test points in the intervals created by the critical points:
$(0, \displaystyle \frac{\pi}{2}),\qquad f'(\frac{\pi}{4})=-0.707...$,
$(\displaystyle \frac{\pi}{2}, \frac{2\pi}{3}),\qquad f'(\frac{5\pi}{8})=-0.0178...$,
$(\displaystyle \frac{2\pi}{3}, \frac{4\pi}{3}),\quad f'(\pi)=1$,
$(\displaystyle \frac{4\pi}{3}$, $2\displaystyle \pi),\quad f'(\frac{15\pi}{8})=-3.937...$
$f':\qquad \stackrel{0}{[} \stackrel{\searrow}{- - } \stackrel{\pi/2}{|} \stackrel{\searrow}{- -} \stackrel{2\pi/3}{|} \stackrel{\nearrow}{++} \stackrel{4\pi/3}{|} \stackrel{\searrow}{- - } \stackrel{2\pi}{]}$
f is increasing on $(\displaystyle \frac{2\pi}{3}, \frac{4\pi}{3})$, decreasing on $(0, \displaystyle \frac{\pi}{2})\cup(\frac{\pi}{2}, \frac{2\pi}{3})\cup(\frac{4\pi}{3}$, $2\pi)$
$(c)$
Local maximum at $x=\displaystyle \frac{4\pi}{3}$ and $x=0$,
local minimum at $x=\displaystyle \frac{2\pi}{3}$ and $ x=2\pi$
