Editing Concave function (section)

==Definition==
A real-valued [[function (mathematics)|function]] <math>f</math> on an [[interval (mathematics)|interval]] (or, more generally, a [[convex set]] in [[vector space]]) is said to be ''concave'' if, for any <math>x</math> and <math>y</math> in the interval and for any <math>\alpha \in [0,1]</math>,<ref>{{cite book |last1=Lenhart |first1=S. |last2=Workman |first2=J. T. |title=Optimal Control Applied to Biological Models |publisher=Chapman & Hall/ CRC |series=Mathematical and Computational Biology Series |year=2007 |isbn=978-1-58488-640-2 }}</ref>

:<math>f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y)</math>

A function is called ''strictly concave'' if

:<math>f((1-\alpha )x+\alpha y) > (1-\alpha ) f(x)+\alpha f(y)</math>

for any <math>\alpha \in (0,1)</math> and <math>x \neq y</math>.

For a function <math>f: \mathbb{R} \to \mathbb{R}</math>, this second definition merely states that for every <math>z</math> strictly between <math>x</math> and <math>y</math>, the point <math>(z, f(z))</math> on the graph of <math>f</math> is above the straight line joining the points <math>(x, f(x))</math> and <math>(y, f(y))</math>.

[[Image:ConcaveDef.png]]

A function <math>f</math> is [[quasiconvex function|quasiconcave]] if the upper contour sets of the function <math>S(a)=\{x: f(x)\geq a\}</math> are convex sets.<ref name=":0">{{Cite book|last=Varian, Hal R.|url=https://www.worldcat.org/oclc/24847759|title=Microeconomic analysis|date=1992|publisher=Norton|isbn=0-393-95735-7|edition=3rd|location=New York|pages=489|oclc=24847759}}</ref>