Concave function

Definition

For a function of a single variable

Suppose $f$ is a function from an interval $(a,b)$ in the real numbers, to the real numbers. We say that $f$ is a concave function on $(a,b)$ if:

${\frac {\partial ^{2}f}{\partial x^{2}}}\leq 0\ \forall \ x\in (a,b)$.

We say that $f$ is a strictly concave function on $(a,b)$ if:

${\frac {\partial ^{2}f}{\partial x^{2}}}<0\ \forall \ x\in (a,b)$.

Concave functions could be of many type:

• Concave functions that are increasing throughout, and in the limit still have positive first derivative.
• Concave functions that are increasing throughout, and in the right limit have zero derivative.
• Concave functions that start off increasing and end up decreasing.
• Concave functions that start off with zero derivative and end up decreasing.
• Concave functions that start off with negative first derivative.

Facts

Negative of convex

A function $f$ is concave if and only if the function $-f$ is a convex function. Similarly, $f$ is a strictly concave function if and only if $-f$ is a strictly convex function.

Closed under addition

A linear combination of concave functions with positive coefficients is again concave.

Inverse function of a concave function is concave

If $f$ is a concave function, and it is invertible, then the inverse function $f^{-1}$ is also a concave function.