Can anyone find a function $f$ with an $x$ such that $f'(x)=0$, $f(x)$ is either a relative max/min and $(x,f(x))$ is an inflection point? In other words, suppose you're using the second derivative test, see that $f''(x)=0$ and then note that it's actually an inflection point. Can you conclude that $(x,f(x))$ is not a relative extremum?
It's easy to find an example where $f''(x)=0$ and there is an extremum--take $f(x)=x^4$, but in that case, there isn't actually an inflection point at 0.
I'm certainly open to the possibility that an inflection point can never be an extremum, though it seems like something I ought to have discovered by now, were it true.