So, I am given that $$f'(x)=x^2(x-2)(x+3)$$ and asked to find where the original function $f$ is increasing and decreasing. The question also asks for the $x$ coordinates of all extreme points and decide whether it's maximum or minimum.
The critical points are $-3,0,2$. So I found that $f$ is increasing on $(-\infty, -3), (2,\infty)$ and decreasing on $(-3,0),(0,2)$.
Using the first derivative test, I got that $f'$ changes sign from $+$ to $-$ around $x=-3$ and so $f$ has a local maximum at $x=3$. $f'$ changes sign from $-$ to $+$ around $x=3$, so $f$ must have a local minimum at $x=2$. However, $f'$ does not change sign around $0$. What am I do make of that?