Encyclopedia > Concave

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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>.
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


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.

See also: convex function.

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