Answer
Suppose that c is the kind of critical point where $f'(c) = 0$. We can sometimes tell whether or not there is a local extremum at $x = c$ by simply looking at the sign of $f''(c)$. If $f''(c) > 0$, we know that $f(c)$ is a local minimum, and if $f''(c) < 0$, we know that $f(c)$ is a local maximum. One useful way to remember this is to also think about concavity, and imagine a nice parabola. If the second derivative is positive at a critical number, then the graph is concave up there, corresponding to a minimum. If the second derivative is negative, then the graph is concave down, corresponding to a maximum. Note that if $f''(c) = 0$, it does not necessarily follow that there isn’t a local extremum at $x = c$. The test doesn’t tell us anything for sure in this case.
