Concave

Concave

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

<math>f\left(\frac{x+y}{2}\right)\geq\frac{f(x)+f(x)}{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, <math>f(x)</math> is concave on <math>[a,b]</math> iff the function <math>-f(x)</math> is convex on every subinterval[?] of <math>[a,b]</math>.

If <math>f(x)</math> is differentiable, then <math>f(x)</math> is concave iff <math>f'(x)</math> is monotone decreasing.

If <math>f(x)</math> is twice-differentiable, then <math>f(x)</math> is concave iff <math>f(x)</math> is negative.