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

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

$f(tx+(1-t)y)geq\; t\; f(x)+(1-t)f(y).$

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

$f(tx\; +\; (1-t)y)\; >\; t\; f(x)\; +\; (1-t)f(y),$

for any t in (0,1) and x ≠ y.

A continuous function on C is concave if and only if

$fleft(frac\{x+y\}2\; right)\; ge\; frac\{f(x)\; +\; f(y)\}2$ .

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 $x\_0$ such that for all f is called quasiconvex if and only if −f is quasiconcave.

Examples

• The function $f(x)=-x^2$ is concave, as its second derivative is always negative.
• Any constant function $f(x)=c$ is both concave and convex.
• The function $f(x)=sin\; x$ is concave on any interval of the form $[2pi\; n,\; 2pi\; n+pi],,$ where $n$ is an integer.

See also

• Convex function
• Quasiconcave function
• Concave polygon
• log-concave