Answer
$ a.\quad$ critical points at $x=\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{3\pi}{4}$, $\displaystyle \frac{5\pi}{4}$, and $\displaystyle \frac{7\pi}{4}$
$ b.\quad$ increasing on $(\displaystyle \frac{\pi}{4}, \frac{3\pi}{4})\cup(\frac{5\pi}{4\pi}, \frac{7\pi}{4})$, decreasing on $(0, \displaystyle \frac{\pi}{4})\cup(\frac{3\pi}{4}, \frac{5\pi}{4})\cup(\frac{7\pi}{4}$, $2\pi)$
$ c.\quad$ Local maxima at $x=0,$ at $x=\displaystyle \frac{3\pi}{4}$ and $x$=$\displaystyle \frac{7\pi}{4}$,
local minima at $x= \displaystyle \frac{\pi}{4}, x \displaystyle \frac{5\pi}{4}$ and $x$=$ 2\pi$
Work Step by Step
$(a)$
$f'$ is defined on $[0,2\pi]$
$f'(x)=0$ for
$\displaystyle \sin x=-\cos x\Rightarrow\quad\tan x=-1\Rightarrow\quad x=\frac{3\pi}{4}$ ,$\displaystyle \frac{7\pi}{4}$
$\displaystyle \sin x=\cos x\Rightarrow\quad\tan x=1\Rightarrow\quad x=\frac{\pi}{4}$ ,$\displaystyle \frac{5\pi}{4}$
Critical points at $x=\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{3\pi}{4}$, $\displaystyle \frac{5\pi}{4}$, and $\displaystyle \frac{7\pi}{4}$
$(b)$
Calculate $f'$ at test points in the intervals created by the critical points:
$(0, \displaystyle \frac{\pi}{4}),\qquad f'(\frac{\pi}{8})=-0.707...$,
$(\displaystyle \frac{\pi}{4}, \frac{3\pi}{4}),\qquad f'(\frac{\pi}{2})=1$,
$(\displaystyle \frac{3\pi}{4}, \frac{5\pi}{4}),\quad f'(\pi)=-1$,
$(\displaystyle \frac{5\pi}{4\pi}, \frac{7\pi}{4}) ,\displaystyle \quad f'(\frac{3\pi}{2})=1$
$(\displaystyle \frac{7\pi}{4}$, $2\pi) ,\displaystyle \quad f'(\frac{15\pi}{8})=-0.707...$,
$f':\qquad \stackrel{0}{[} \stackrel{\searrow}{- - -} \stackrel{\pi/4}{|} \stackrel{\nearrow}{+++} \stackrel{3\pi/4}{|} \stackrel{\searrow}{- - -} \stackrel{5\pi/4}{|} \stackrel{\nearrow}{+++} \stackrel{7\pi/4}{|}$ $\stackrel{\searrow}{- - -} \stackrel{2\pi}{]}$
f is increasing on $(\displaystyle \frac{\pi}{4}, \frac{3\pi}{4})\cup(\frac{5\pi}{4\pi}, \frac{7\pi}{4})$, decreasing on $(0, \displaystyle \frac{\pi}{4})\cup(\frac{3\pi}{4}, \frac{5\pi}{4})\cup(\frac{7\pi}{4}$, $2\pi)$
$(c)$
Local maxima at $x=0,$ at $x=\displaystyle \frac{3\pi}{4}$ and $x$=$\displaystyle \frac{7\pi}{4}$,
local minima at $x= \displaystyle \frac{\pi}{4}, x \displaystyle \frac{5\pi}{4}$ and $x$=$ 2\pi$
