Prove that $f(x) ≥ f(x^∗) + α||x − x^∗||\iff 0 ∈ \text{int}\ ∂f (x^∗)$

Question

Assume that $f$ is a convex function, and $x^*$ is the minimizer of the function.

Prove that the inequality $f(x) ≥ f(x^*) + α||x − x^*|| (α > 0)$ holds if, and only if, $0 ∈ \text{int}\ ∂f(x^*)$.

I have been stuck on this problem from over a week now, and was hoping to get some help from you guys.