# Second Derivative of Concave Real Function is Non-Positive

Jump to navigation
Jump to search

## Contents

## 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 $f'' \le 0$.

$\blacksquare$

## Also see

- Second Derivative of Convex Real Function is Non-Negative
- Second Derivative of Strictly Convex Real Function is Strictly Positive

## Sources

- For a video presentation of the contents of this page, visit the Khan Academy.