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If your function was strictly convex and $C^3$ (let's say on a compact interval $I$) and you had bounds on the third derivative there, it might be possible to prove its convexity on $I$ by evaluating it at sufficiently many points of $I$. If not, it's impossible to distinguish any convex function from one whose second derivative has a narrow "blip" taking it below zero between some of the points where you evaluated it.