 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
Definition
A realvalued 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]}
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 "nonincreasing", rather than "strictly decreasing", and thus allows zero slopes.)
For a twicedifferentiable 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 twicedifferentiable, then f(x) is concave if and only if f ′′(x) is nonpositive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = x^{4}.
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,
Examples
 The functions f(x) = − x^{2} and are concave, as the second derivative is always negative.
 Any linear function f(x) = ax + b is both concave and convex.
 The function f(x) = sin(x) is concave on the interval [0,π].
 The function log  B  , where  B  is the determinant of a nonnegativedefinite matrix B, is concave.^{[3]}
 Practical application: rays bending in Computation of radiowave attenuation in the atmosphere.
See also
 Concave polygon
 Convex function
 Jensen's inequality
 Logarithmically concave function
 Quasiconcave function
References
 ^ Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 496
 ^ Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 489
 ^ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM journal on matrix analysis and applications 9 (3): 384–392.
 Crouzeix, J.P. (2008). "Quasiconcavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375. http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008.
 Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0470183527.
Categories: Mathematical analysis
 Types of functions
 Convex analysis
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