I am not sure what the relevant theorems for this problem are. I have been searching through Rudin for some hints, but I have come up short. This is an example question for an exam, so not homework.
Can anyone point me in the right direction? Thanks.
A function $f: \mathbb{R}^n \to \mathbb{R}$ is called convex if $f$ satifies $ f(\alpha x + (1 - \alpha)y) \le \alpha f(x) + (1 - \alpha)f(y) \quad \forall x, y \in \mathbb{R}^n,\ 0 \le \alpha \le 1.$ Assume that $f$ is continuously differentiable and that for some constant $c > 0$, the gradient $(\nabla f(x) - \nabla f(y)) \cdot (x - y) \ge c(x - y) \cdot (x - y), \quad \forall x, y \in \mathbb{R}^n,$ where $\cdot$ denotes the dot product. Show that $f$ is convex.