# 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 $\openint a b$.

Then $f$ is concave on $\openint a b$ if and only if its second derivative $f* \le 0$ on $\openint a b$.*

## 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 $f* \le 0$.*

$\blacksquare$