Editing Convex function (section)

[[Image:ConvexFunction.svg|thumb|300px|right|Convex function on an [[interval (mathematics)|interval]].]]{{Use American English|date = March 2019}}
{{Short description|Real function with secant line between points above the graph itself}}
[[Image:Epigraph convex.svg|right|thumb|300px|A function (in black) is convex if and only if the region above its [[Graph of a function|graph]] (in green) is a [[convex set]].]] 
[[Image:Grafico 3d x2+xy+y2.png|right|300px|thumb|A graph of the [[polynomial#Number of variables|bivariate]] convex function {{nowrap| ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>}}.]]
[[File:Convex vs. Not-convex.jpg|thumb|right|300px|Convex vs. Not convex]]

In [[mathematics]], a [[real-valued function]] is called '''convex''' if the [[line segment]] between any two distinct points on the [[graph of a function|graph of the function]] lies above or on the graph between the two points. Equivalently, a function is convex if its [[epigraph (mathematics)|''epigraph'']] (the set of points on or above the graph of the function) is a [[convex set]]. 
In simple terms, a convex function graph is shaped like a cup <math>\cup</math> (or a straight line like a linear function), while a [[concave function]]'s graph is shaped like a cap <math>\cap</math>.

A twice-[[differentiable function|differentiable]] function of a single variable is convex [[if and only if]] its [[second derivative]] is nonnegative on its entire [[domain of a function|domain]].<ref>{{Cite web|url=https://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf |title=Lecture Notes 2|website=www.stat.cmu.edu|access-date=3 March 2017}}</ref> Well-known examples of convex functions of a single variable include a [[linear function]] <math>f(x) = cx</math> (where <math>c</math> is a [[real number]]), a [[quadratic function]] <math>cx^2</math> (<math>c</math> as a nonnegative real number) and an [[exponential function]] <math>ce^x</math> (<math>c</math> as a nonnegative real number).

Convex functions play an important role in many areas of mathematics. They are especially important in the study of [[optimization]] problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an [[open set]] has no more than one [[maximum and minimum|minimum]]. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the [[calculus of variations]]. In [[probability theory]], a convex function applied to the [[expected value]] of a [[random variable]] is always bounded above by the expected value of the convex function of the random variable.  This result, known as [[Jensen's inequality]], can be used to deduce [[inequality (mathematics)|inequalities]] such as the [[inequality of arithmetic and geometric means|arithmetic&ndash;geometric mean inequality]] and [[Hölder's inequality]].