Answer
Relative maximum: $ x=\frac{5 \pi}{4}, x=\frac{\pi}{4}, x=\frac{3 \pi}{4},x=$$\frac{7 \pi}{4}$
Relative minimum: $x=\pi, x=\frac{3 \pi}{2} , x=\frac{\pi}{2}$
Work Step by Step
First derivative
\[
\frac{2|\sin 2 x| \cos 2 x}{\sin 2 x}=f^{\prime}(x)
\]
Critical points (zeros of first derivative or point where first derivative does not exist)
\[
x=\pi, x=\frac{5 \pi}{4}, x=\frac{3 \pi}{2}, x=\frac{\pi}{4}, x=\frac{\pi}{2}, x=\frac{3 \pi}{4},x=\frac{7 \pi}{4}
\]
Second derivative
\[
-4|\sin 2 x|=f^{\prime \prime}(x)
\]
Values of the second derivative at the critical point:
$f^{\prime \prime}\left(\frac{\pi}{4}\right)=-4<0$
$f^{\prime \prime}\left(\frac{\pi}{2}\right)=0$
$f^{\prime \prime}\left(\frac{3 \pi}{4}\right)=-4<0$
$f^{\prime \prime}(\pi)=0$
$f^{\prime \prime}\left(\frac{5 \pi}{4}\right)=-4<0$
$f^{\prime \prime}\left(\frac{3 \pi}{2}\right)=0$
$f^{\prime \prime}\left(\frac{7 \pi}{4}\right)=-4<0$
Relative maximum: $ x=\frac{5 \pi}{4}, x=\frac{\pi}{4}, x=\frac{3 \pi}{4},x=$$\frac{7 \pi}{4}$
Relative minimum: $x=\pi, x=\frac{3 \pi}{2} , x=\frac{\pi}{2}$
