Second Derivative of Convex Real Function is Non-Negative

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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 Real Function is Convex iff Derivative is Increasing, $f$ is convex if and only if $f'$ is increasing.

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

$\blacksquare$


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