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# Concave

In mathematics, a function $f(x)$ is said to be concave on an interval $[a,b]$ if, for all x,y in $[a,b]$.
$f\left(\frac{x+y}{2}\right)\geq\frac{f(x)+f(x)}{2}$
This is equivalent to
$\forall t\in[0,1],\ \ f(tx + (1-t)y) \geq tf(x) + (1-t)f(y).$

Additionally, $f(x)$ is strictly concave if

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

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.

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