Answer
See explanation.
Work Step by Step
Points where the tangent line is horizontal (and thus the derivative zero) are critical points, meaning the function is not differentiable.
\[
x=\pm \frac{3}{2} \text { and } x=0
\]
Using first derivative test
$f^{\prime}(x)<0$ on the left and $f^{\prime}(x)>0$ on the right of $x=-\frac{3}{2},$ so $x=-\frac{3}{2}$ is a relative minimum.
Using the first derivative test
$f^{\prime}(x)>0$ on the left and $f^{\prime}(x)<0$ on the right of $x=0,$ so $x=0$ is a relative maximum.
Using the first derivative test
$f^{\prime}(x)<0$ on the left and $f^{\prime}(x)>0$ on the right of $x=\frac{3}{2},$ so $x=\frac{3}{2}$ is a relative minimum.
