Concave function

In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements.

Equivalently, a concave function is any function for which the hypograph is convex.

The class of concave functions is in a sense the opposite of the class of convex functions.

A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

A real-valued function

on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any

,[1] A function is called strictly concave if additionally any

, this second definition merely states that for every

is above the straight line joining the points

is quasiconcave if the upper contour sets of the function

are convex sets.

A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive