Second Derivative of Convex Real Function is Non-Negative

Theorem
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.

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

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

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

Also see

 * Twice Differentiable Real Function with Positive Second Derivative is Strictly Convex


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