I am reading the Wikipedia of Convex function. in it, it is written:
if $f$ is a convex function then $$ f(x) \ge f(y) + f'(y)(x-y)$$ for all $x$ and $y$ in the interval. In particular, if $f ′(c) = 0$, then $c$ is a global minimum of $f(x)$.
I fully understand this, but I'm wondering if it is possible to prove that if $c$ if the point of the minimum then $f ′(c) = 0$ just by using convexity. I think it is possible but until now I could not prove that.