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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).
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