# Second Derivative of Strictly Convex Real Function is Strictly Positive

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## Theorem

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

Then $f$ is strictly convex on $\openint a b$ if and only if 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 if and only if $f'$ is strictly increasing.

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

$\blacksquare$