Second Derivative of Concave Real Function is Non-Positive

Theorem
Let $f$ be a differentiable real function on some interval $\mathbb I$. If $f''\left(x\right) < 0$ for all $x$ on the interval, $f$ is concave down.

Proof
By Derivative of Monotone Function, If $f''\left(x\right) < 0$, then $f'$ is strictly decreasing.

The result follows from the definition of downward-concavity for differentiable functions.

Also see

 * Second Derivative of Concave Upward Function