Concave function

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.

Definition

A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any x and y in the interval and for any t in [0,1],

${\displaystyle f((1-t)x+(t)y)\geq (1-t)f(x)+(t)f(y).}$

A function is called strictly concave if

${\displaystyle f((1-t)x+(t)y)>(1-t)f(x)+(t)f(y)\,}$

for any t in (0,1) and xy.

For a function f:RR, this definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(y) ).

A function f(x) is quasiconcave if the upper contour sets of the function ${\displaystyle S(a)=\{x:f(x)\geq a\}}$ are convex sets.Template:Sfn

Properties

A function f(x) is concave over a convex set if and only if the function −f(x) is a convex function over the set.

A differentiable function f is concave on an interval if its derivative function f ′ is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means non-increasing, rather than strictly decreasing, and thus allows zero slopes.)

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; if f ′′(x) is negative, then the graph is concave. 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.

If f(x) is twice-differentiable, then f(x) is concave if and only if 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.

If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:

${\displaystyle f(y)\leq f(x)+f'(x)[y-x]}$Template:Sfn

A continuous function on C is concave if and only if for any x and y in C

${\displaystyle f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}}$

If a function f is concave, and f(0) ≥ 0, then f is subadditive. Proof:

Notes

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References

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