Concavity

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Concavity is a geometric term which describes a curve. In calculus, a graph is concave upward if the derivative, f '(x) (of the function, f(x) being graphed) is increasing upon an interval; a graph is concave downward if the derivative is decreasing. 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 pointss.

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 is said to be concave on an interval if, for all x,y in .

This is equivalent to

Additionally, is strictly concave if

Equivalently, is concave on iff the function is convex on every subinterval of .

If is differentiable, then is concave iff is monotone decreasing.

If is twice-differentiable, then is concave iff 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