Real Function with Strictly 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 $\openint a b$ 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.

$\blacksquare$

Also see

• Second Derivative of Convex Real Function is Non-Negative
• Real Function with Strictly Positive Second Derivative is Strictly Convex

• Second Derivative of Concave Real Function is Non-Positive