When looking over true/false questions on previous midterms, one of my conscientious students said:
"If f is defined on an open interval containing c, f'(c)=0, and f''(c)>0, then c is a local min of f"
was false because one of the hypotheses for the second derivative test (at least in Stewart) is that the second derivative is continuous in a neighborhood of c.
Can anyone think of a counterexample for this statement (in one real variable)? It's apparently been too long since I've taken an analysis class to come up with something clever.