# Second Derivative of Strictly Convex Real Function is Strictly Positive

Jump to navigation
Jump to search

## 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 $f'' > 0$.

$\blacksquare$