# Second Derivative of Strictly Concave Real Function is Strictly 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 strictly concave on $\left({a \,.\,.\, b}\right)$ if and only if its second derivative $f'' < 0$ on $\left({a \,.\,.\, b}\right)$.

## Proof

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

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

$\blacksquare$