The First Derivative Test
Remember that critical numbers are the only places where a function
can have a local maximum or minimum (aside from endpoints), and are the
only places where $f'(x)$ can change sign. These are related by
The First Derivative Test:
Let $c$ be a critical number for a continuous function $f$
 If $f'(x)$ changes from positive to negative at $c$, then $f(c)$
is a local maximum.
 If $f'(x)$ changes from negative to positive at $c$, then $f(c)$
is a local minimum.
 If $f'(x)$ does not change sign at $c$, then $f(c)$ is neither a
local maximum or minimum.

This should jibe with your common sense. A local maximum is where you
stop going up and start coming down. A local minimum is where you stop going
down and start coming up. If you flatten out and then resume going in
the direction you were already heading, you're at a critical point but
not at a maximum or minimum. Once you make a sign chart for $f'(x)$,
as we outlined previously, then you know where all the local maxima and
minima are.
Example: Find the maxima and minima of $f(x)=x^33x^2$.
Solution:
We already determined that
$f'(x)$ is positive on $(\infty,0)$, negative on $(0,2)$ and positive
on $(2,\infty)$. Since $f'$ goes from positive to negative at $x=0$, there
is a local maximum at $x=0$. Since $f'$ goes from negative to positive
at $x=2$, there is a local minimum at $x=2$. 
