# 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.*

$\blacksquare$