Remember to investigate all extrema CANDIDATES. That means, find all values of x where f'(x) = 0 AND f '(x) DNE.
After all, a max or min could exist at a cusp... But remember, sometimes these candidates turn out to be inflection points instead.
When looking for points of inflection, remember to consider where concavity changes. This could happen where f ''(x) = 0
or where f ''(x) = DNE.