Concavity

geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. In calculus, a differentiable function is concave upward if the derivative, f ′(x) (of the function, f(x) being graphed) is increasing upon an interval; a twice-differentiable function is concave downward if the derivative is decreasing.

Contrary to the impression one may get from a calculus course, differentiability is not essential to these concepts; see convex.

In other words, if the second derivative, f ''(x), is positive (or, if the acceleration is positive); then, the graph is concave upward; if the second derivative is negative; then, the graph is concave downward. Points where concavity changes are inflection points.

The "bottom" of a concave downward slope will have a point known as the minimal extremum; the "apex" of a concave upward slope will have a point known as the maximal extremum.

In mathematics, a function f(x) is said to be concave on an interval [a, b] if, for all x,y in [a, b],

<math>f\left(\frac{x+y}{2}\right)\geq\frac{f(x)+f(y)}{2}<math>

This is equivalent to

<math>\forall t\in[0,1],\ \ f(tx + (1-t)y) \geq tf(x) + (1-t)f(y).<math>

Additionally, <math>f(x)<math> is strictly concave if

<math>f\left(\frac{x+y}{2}\right)>\frac{f(x)+f(y)}{2}.<math>

Equivalently, f(x) is concave on [a, b] iff the function −f(x) is convex on every subinterval of [a, b].

If f(x) is differentiable, then f(x) is concave iff f ′(x) is monotone decreasing.

If f(x) is twice-differentiable, then f(x) is concave iff f ′′(x) is negative.

Concave polygons

In a concave polygon, some angle will be greater than 180°. The extension at that vertx of the line segment making up a side will pass through the interior of the polygon.

A concave polygon is also called re-entrant.

See also: convex