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 $\openint a b$ such that:


 * $\map {f''} x < 0$ for each $x \in \openint a b$.

Then $f$ is strictly concave on $\openint a b$ its second derivative $f'' < 0$ on $\openint a b$.

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

Since $f'' < 0$, we have that $f'$ is strictly decreasing from Real Function with Strictly Negative Derivative is Strictly Decreasing.

Also see

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


 * Second Derivative of Concave Real Function is Non-Positive