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 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
• Real Function with Strictly Positive Second Derivative is Strictly Convex

• Real Function with Strictly Negative Second Derivative is Strictly Concave