# Second Derivative of Concave Real Function is Non-Positive

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

$\blacksquare$