Second Derivative of Concave Real Function is Non-Positive

Theorem
Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Then $f$ is convex on $\left({a \,.\,.\, b}\right)$ iff its second derivative $f'' \le 0$ on $\left({a \,.\,.\, b}\right)$.

Proof
From Derivative of Concave Real Function is Decreasing, $f$ is concave iff $f'$ is decreasing.

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

Also see

 * Second Derivative of Convex Real Function