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[[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 points on the [[graph of a function|graph of the function]] lies above 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]]. A twice-differentiable function of a single variable is convex [[if and only if]] its second derivative is nonnegative on its entire ___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 the [[quadratic function]] <math>x^2</math> and the [[exponential function]] <math>e^x</math>. In simple terms, a convex function refers to a function whose graph is shaped like a cup <math>\cup</math>, while a [[concave function]]'s graph is shaped like a cap <math>\cap</math>.
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