Concave function

In mathematics, a real-valued function f defined on an interval (or on any concave subset 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

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

In other words, a function is concave if and only if its epigraph (the set of points lying on or above the graph) is a non-convex set.

A function that is concave is often synonymously called concave downward, and a function that is convex is often synonymously called concave upwards.

A function is called strictly concave if

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

for any t in (0,1) and x ≠ y.

The opposite concept of concave function is convex function.

A continuous function on C is concave if and only if

$f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}$ .

for any x and y in C. Equivalently, f(x) is concave on [a, b] if and only if the function −f(x) is convex on every subinterval of [a, b].

A differentiable function f is concave on an interval if its derivative function f ′ is monotone 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 $x<x_{0}$, $f(x)$ is non-decreasing while for all $x>x_{0}$ it is non-increasing. $x_{0}$ can also be $+(-)\infty$, making the function non-decreasing (non-increasing) for all $x$. The opposite of quasiconcave is quasiconvex.

See also

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

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