Answer
If $f'(c) = 0$ and $f''(c) > 0$, $f(c)$ is a local minimum; if $f'(c) = 0$ and $f''(c) < 0$, $f(c)$ is a local maximum.
Explanation
If $f''(c) = 0$, the test is inconclusive and higher derivatives or sign changes of $f'(x)$ must be examined.
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What is the condition for an extremum (local minimum or maximum) of a function $f(x)$ using second derivatives?
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