Concave function

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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} \le 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.