Twice Differentiable Real Function with Positive Second Derivative is Strictly Convex

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

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

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

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

Also see

 * Second Derivative of Convex Real Function is Non-Negative


 * Second Derivative of Concave Real Function is Non-Positive
 * Second Derivative of Strictly Concave Real Function is Strictly Negative