Twice Differentiable Real Function with Negative Second Derivative is Strictly Concave

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 $\left({a \,.\,.\, b}\right)$ iff 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 iff $f'$ is strictly decreasing.

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

Also see

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


 * Second Derivative of Concave Real Function is Non-Positive