Real Function with Strictly Negative Second Derivative is Strictly Concave
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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 $\openint a b$ if and only if 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 if and only if $f'$ is strictly decreasing.
Since $f < 0$, we have that $f'$ is strictly decreasing from Real Function with Strictly Negative Derivative is Strictly Decreasing.
$\blacksquare$