Concave function

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.

Contents

 [hide] 

• 1 Definition
• 2 Properties
• 3 Examples
• 4 See also
• 5 Notes
• 6 References

[edit]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],

A function is called strictly concave if

for any t in (0,1) and x ≠ y.

For a function f:R→R, 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  are convex sets.^[1]

[edit]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

^[2]

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

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

• since f is concave, let y = 0, 
• 

[edit]Examples

• The functions  and  are concave, as the second derivative is always negative.
• Any linear function  is both concave and convex.
• The function  is concave on the interval .
• The function , where  is the determinant of a nonnegative-definite matrix B, is concave.^[3]
• Practical example: rays bending in Computation of radiowave attenuation in the atmosphere.

[edit]See also

• Concave polygon
• Convex function
• Jensen's inequality
• Logarithmically concave function
• Quasiconcave function

[edit]Notes

1. ^ Varian 1992, p. 496.
2. ^ Varian 1992, p. 489.
3. ^ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM journal on matrix analysis and applications 9 (3): 384–392.

[edit]References

• Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375.
• Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0-470-18352-7.
• Varian, Hal R. (1992). Microeconomic Analysis (Third ed.). W.W. Norton and Company.