Concave Function

analysis

Definition

Concave Function

A function is called concave on an interval if for all and all :

Geometrically, this means that the line segment (secant) between any two points on the graph of the function lies below or on the graph.

Properties

• Relation to Convexity: A function is concave if and only if is a convex function.
• Second Derivative Test: If is twice differentiable on an open interval, then is concave if and only if its second derivative is non-positive:.
• Minima and Maxima: Any local maximum of a concave function is also an absolute maximum.
• Jensen’s Inequality: For a concave function ,.