Second Derivative of Concave Real Function is Non-Positive

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


Proof

From Real Function is Concave iff Derivative is Decreasing, $f$ is concave if and only if $f'$ is decreasing.

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

$\blacksquare$


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