Answer
The First Derivative Test can be used to tell whether or not a critical point is a local maximum or minimum, as follows: If (c, f(c)) is a critical point, we investigate the sign of $f'$ for points that are just to the left and just to the right of $c$. If the sign of $f'$ changes from positive to negative, then $f$ is changing from increasing to decreasing at $c$, so there is a local maximum at $c$. If the signs of $f'$ are changing from negative to positive, then $f$ is changing from decreasing to increasing at $c$, so there is a local minimum at $c$. If the signs of $f'$ are the same on either side of $c$, then there is neither kind of local extremum at $x = c$.
Note that if we find all of the critical points of $f$, and if the domain of $f$ is an interval or union of intervals, then the critical points naturally divide up the domain into intervals on which we can check the sign of $f'$ and look for places where the sign changes.
Work Step by Step
See above.
