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m The first sentence was incorrect, if the line segment had to be above the graph then f(x) = cx is not convex. I believe the previous author was in a hurry, this is a minor edit. Please search for ORF523_S16_Lec7_gh.pdf to see an introduction to the topic if this change feels strange. |
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* The [[absolute value]] function <math>f(x)=|x|</math> is convex (as reflected in the [[triangle inequality]]), even though it does not have a derivative at the point <math>x = 0.</math> It is not strictly convex.
* The function <math>f(x)=|x|^p</math> for <math>p \ge 1</math> is convex.
* The [[exponential function]] <math>f(x)=e^x</math> is convex. It is also strictly convex, since <math>f''(x)=e^x >0 </math>, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function <math>g(x) = e^{f(x)}</math> is [[Logarithmically convex function|logarithmically convex]] if <math>f</math> is a convex function. The term "superconvex" is sometimes used instead.<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | doi = 10.1093/qmath/12.1.283 | title = A Convexity Property of Positive Matrices | journal = The Quarterly Journal of Mathematics | volume = 12 | pages = 283–284 | year = 1961 | bibcode = 1961QJMat..12..283K }}</ref>
* The function <math>f</math> with ___domain [0,1] defined by <math>f(0) = f(1) = 1, f(x) = 0</math> for <math>0 < x < 1</math> is convex; it is continuous on the open interval <math>(0, 1),</math> but not continuous at 0 and 1.
* The function <math>x^3</math> has second derivative <math>6 x</math>; thus it is convex on the set where <math>x \geq 0</math> and [[concave function|concave]] on the set where <math>x \leq 0.</math>
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