Let $f:\mathbb{R}^d\to \mathbb{R}$ be a differentiable convex function and $S$ by a closed convex set. If $x \in S$ in the minimizer of $f$ over $S$, I want to prove that, for any $z \in S$, $$ \langle \nabla f(x), z-x \rangle \geq 0. $$
Can anyone provide a hint or suggest a method about how to approach this problem?