Locally convex or concave function and differentiable function

Question

If we define a function $f:\Bbb R\to\Bbb R$, locally convex at a point $x$, if there exist an interval around $x$ in which $f$ is convex,

will $f$ differentiable imply $f$ is locally convex or locally concave?

Answer 1

$f = x^3 $ is differentiable at $x=0$ but nether locally convex nor concave around $x=0$

P.S: If you are eager to find conditions implying local convexity or concavity, one sufficient condition is $f$ being $C^2$ then for every $x \in \Bbb R$ such that $f'' (x) \neq 0 $ then $f$ is either locally convex or concave around $x$. But be carful that this cannot be generalized to functions with several arguments unfortunately. That's the reason why many optimization algorithms do not converge in multidimensional spaces unless under some conditions.