Concavity

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In mathematical analysis, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions.

Concave functions

In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. Similarly, a differentiable function f is concave on an interval if its derivative function f ′ is decreasing on that interval: a concave function has a decreasing slope.

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

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 (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward). 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.

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

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

A continuous function on C is concave if and only if

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

for any x and y in C.

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 twice-differentiable, then f(x) is concave iff 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 iff there is an <math>x_0</math> such that for all <math>x<x_0, f(x)</math> is non-decreasing while for all <math>x>x_0</math> it is non-increasing. <math>x_0</math> can also be <math>+(-) \infty</math>, making the function non-decreasing (non-increasing) for all <math>x</math>. The opposite of quasiconcave is quasiconvex.

Concave polygons

In a concave polygon, some interior angle is greater than 180°. The extension at that vertex of the line segment making up a side passes through the interior of the polygon.

A concave polygon is often called re-entrant polygon (but in some cases the latter term has a different meaning).

See also

convex