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 $\openint a b$.

Then $f$ is strictly convex on $\openint a b$ its second derivative $f'' > 0$ on $\openint a b$.

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

From Derivative of Monotone Function, $f'$ is strictly increasing 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