Concave function

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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.

1 Definition
2 Properties
3 Examples
4 See also

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.

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

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

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 $x < x 0$ , $f (x)$ is non-decreasing while for all $x > x 0$ it is non-increasing. $x 0$ can also be $\pm \infty$ , 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 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 $[2\pi n, 2\pi n+\pi],\,$ where $n$ is an integer.

Concave function

Contents

Definition

Properties

Examples

See also