Real Function is Strictly Concave iff Derivative is Strictly Decreasing

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Theorem

Let $f$ be a real function which is differentiable on the open interval $\left({a \,.\,.\, b}\right)$.


Then $f$ is strictly concave on $\left({a \,.\,.\, b}\right)$ iff its derivative $f'$ is strictly decreasing on $\left({a \,.\,.\, b}\right)$.


Proof

Necessary Condition

Let $f$ be strictly concave on $\left({a \,.\,.\, b}\right)$.

Let $x_1, x_2, x_3, x_4 \in \left({a \,.\,.\, b}\right): x_1 < x_2 < x_3 < x_4$.

By the definition of strictly concave:

$\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2} > \dfrac {f \left({x_4}\right) - f \left({x_3}\right)} {x_4 - x_3}$

Ignore the middle term and let $x_2 \to x_1^+$ and $x_3 \to x_4^-$.

Thus:

$f' \left({x_1}\right) > f' \left({x_4}\right)$

Hence $f'$ is strictly decreasing on $\left({a \,.\,.\, b}\right)$.

$\Box$


Sufficient Condition

Let $f'$ be strictly decreasing on $\left({a \,.\,.\, b}\right)$.

Let $x_1, x_2, x_3 \in \left({a \,.\,.\, b}\right): x_1 < x_2 < x_3$.

By the Mean Value Theorem:

$\exists \xi: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} = f' \left({\xi}\right)$
$\exists \eta: \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2} = f' \left({\eta}\right)$

where $x_1 < \xi < x_2 < \eta < x_3$.

Since $f'$ is strictly decreasing:

$f' \left({\xi}\right) > f' \left({\eta}\right)$

Thus:

$\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

Hence $f$ is strictly concave by definition.

$\blacksquare$


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