I would like to know if every sufficiently differentiable function is convex near a local minimum. The background to my question is that I became curious if one could motivate the usefulness of convex optimization techniques by saying that at least locally they work for all continuous functions.

Unfortunately I discovered there are $C^1$-functions which have minima where they are not locally convex. But my construction involves the use of everywhere non-differentiable continuous functions so I started to wonder if perhaps $C^2$-functions are convex near a local minimum.