Answer
Relative maximum: $x=\pi$
Relative minimum: $x=\frac{3 \pi}{2}$ and $x=\frac{ \pi}{2}$
Work Step by Step
First derivative
\[
f^{\prime}(x)=-2 \sin x \cos x
\]
Critical points (zeros of first derivative or point where first derivative does not exist):
\[
x=\frac{3 \pi}{2}, x=\pi \text { and } x=\frac{ \pi}{2}
\]
Second derivative
\[
f^{\prime \prime}(x)=-2 \cos ^{2} x+2 \sin ^{2} x
\]
Values of second derivative at critical points:
\[
f^{\prime \prime}\left(\frac{\pi}{2}\right)=2>0
\]
$f^{\prime \prime}(\pi)=-2<0$
\[
f^{\prime \prime}\left(\frac{3 \pi}{2}\right)=1>0
\]
Relative minimum: $x=\frac{\pi}{2}$ and $x=\frac{3 \pi}{2}$
Relative maximum: $x=\pi$
