# Are all critical points either inflection points, local minimum or local maximum?

Question:

• Let $$f$$ be a differentiable function. If the point c is a critical number, then either it is a local maximum, or local minimum, or an inflection point. $$T/F$$ ?

My opinion:

• If c is a critical point then f'(c)=0 or undefined. So it may local maximum and local minimum.

• If f '(c)=∞ then c is inflection point at the same time and if f '(c)=0 it may inflection point again.

• But i can't find instance disproves this thesis.

$$f(x)=\begin{cases} x^2\sin\left(\frac{1}{x}\right) & x \neq 0\\ 0 & x=0 \end{cases}$$
at $$x=0$$?
$$f$$ is differentiable on $$\mathbb R$$ but $$0$$ is not a minimum, not a maximum and not an inflexion point.
• @Raffaele I imagine that the example is used in may Calculus courses! $f$ is an easy differentiable locally oscillating map! – mathcounterexamples.net Jan 9 at 16:16