A convex region is where the second derivative of a function keeps the same sign.
Consider the function $$ f(x) = sin(x), x \in [0, 2\pi], $$
what are the avalaible algorithm to estimate the number of convex region numerically?
Of course, the convex region are $[0,\pi]$ and $[\pi, 2\pi]$, because $$ f''(x) = -sin(x). $$ But, if the function was a black-box, numerical investigation would be the only resort.