# Extreme values and monotonicity

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.

Attempt:
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?

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If $f'$ doesn't change sign, it's a point of inflection, i.e. not a local min/max.