Second Derivative of Concave Real Function is Non-Positive
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Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.
Then $f$ is concave on $\openint a b$ if and only if its second derivative $f'' \le 0$ on $\openint a b$.
From Real Function is Concave iff Derivative is Decreasing, $f$ is concave if and only if $f'$ is decreasing.
From Derivative of Monotone Function, $f'$ is decreasing if and only if its second derivative $f'' \le 0$.
- Second Derivative of Convex Real Function is Non-Negative
- Twice Differentiable Real Function with Positive Second Derivative is Strictly Convex