A student claims that

If a curve $y=f(x)$ satisfies $\frac{d^{2}y}{d{x}^2}=0$ at some point, then it has a point of inflection at that point.

Explain, with the aid of a diagram, why

is a counterexample to this claim.

You can show that the curve has a local minimum point at $x=0$ by considering the fact that the curve is symmetrical about the $y$ axis (because $f(x)=f(-x)$, i.e. the function $x^4$ is even).