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Concave function
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In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.
ally, a real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we have
Also, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b].
A function is called strictly concave if
for any t in (0,1) and x ? y.
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 continuous function on C is concave if and only if
for any x and y in C.
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.
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Encyclopedia
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.
Definition
Formally, a real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we have
Also, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b].
A function is called strictly concave if
for any t in (0,1) and x ? y.
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 continuous function on C is concave if and only if
for any x and y in C.
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.)
Properties
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.
A function is called quasiconcave if and only if there is an such that for all , is non-decreasing while for all it is non-increasing. can also be , making the function non-decreasing (non-increasing) for all . Also, a function f is called quasiconvex if and only if −f is quasiconcave.
Examples
- The function is concave, as its second derivative is always negative.
- Any constant function is both concave and convex.
- The function is concave on any interval of the form where is an integer.
See also
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