Second Derivative of Convex Real Function is Non-Negative

Theorem
Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Then $f$ is convex on $\left({a \,.\,.\, b}\right)$ iff its second derivative $f'' \ge 0$ on $\left({a \,.\,.\, b}\right)$.

Proof
From Derivative of Convex Real Function is Increasing, $f$ is convex iff $f'$ is increasing.

From Derivative of Monotone Function, $f'$ is increasing iff its second derivative $f'' \ge 0$.

Also see

 * Second Derivative of Concave Real Function