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.