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 $\openint a b$ its second derivative $f'' \le 0$ on $\openint a b$.

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

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

Also see

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


 * Twice Differentiable Real Function with Negative Second Derivative is Strictly Concave