Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.

Definition

Formally, a real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we have

Also, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b].

A function is called strictly concave if

for any t in (0,1) and x ≠ y.

This definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(y) ).

A continuous function on C is concave if and only if

for any x and y in C.

A differentiable function f is concave on an interval if its derivative function f ′ is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means "non-increasing", rather than "strictly decreasing", and thus allows zero slopes.)

Properties

For a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave. Points where concavity changes are inflection points.

If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.

If f(x) is twice-differentiable, then f(x) is concave if and only if f ′′(x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x⁴.

A function is called quasiconcave if and only if there is an x₀ such that for all x < x₀, f(x) is non-decreasing while for all x > x₀ it is non-increasing. x₀ can also be , making the function non-decreasing (non-increasing) for all x. Also, a function f is called quasiconvex if and only if −f is quasiconcave.

Examples

• The functions f(x) = − x² and are concave, as the second derivative is always negative.
• Any linear function f(x) = ax + b is both concave and convex.
• The function f(x) = sin(x) is concave on any interval of the form where n is an integer.

See also

• Convex function
• Quasiconcave function
• Concave polygon
• log-concave
• Jensen's inequality