## Continuous Optimization: Proof of Convexity

Q. Let f be smooth convex function defined on convex domain. Using the definition of convexity and the directional derivative, show that

(gradient f(y) - gradient f(x)) * (y - x) >= 0.

I already know that f(y) >= f(x) + gradient f(x) * (y - x), but how can I use this to solve the aforementioned proof. I would greatly appreciate any guidance! Thanks.