# 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 $\left({a \,.\,.\, b}\right)$ if and only if its second derivative $f'' \ge 0$ on $\left({a \,.\,.\, b}\right)$.

## Proof

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

$\blacksquare$