Editing Convex function (section)

==Examples==
===Functions of one variable===
* The function <math>f(x)=x^2</math> has <math>f''(x)=2>0</math>, so {{mvar|f}} is a convex function.  It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
* The function <math>f(x)=x^4</math> has <math>f''(x)=12x^2\ge 0</math>, so {{mvar|f}} is a convex function.  It is strictly convex, even though the second derivative is not strictly positive at all points.  It is not strongly convex.
* 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&nbsp;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>
* Examples of functions that are [[Monotonic function|monotonically increasing]] but not convex include <math>f(x)=\sqrt{x}</math> and <math>g(x)=\log x</math>.
* Examples of functions that are convex but not [[Monotonic function|monotonically increasing]] include <math>h(x)= x^2</math> and <math>k(x)=-x</math>.
* The function <math>f(x) = \tfrac{1}{x}</math> has <math>f''(x)=\tfrac{2}{x^3}</math> which is greater than 0 if <math>x > 0</math> so <math>f(x)</math> is convex on the interval <math>(0, \infty)</math>. It is concave on the interval <math>(-\infty, 0)</math>.
* The function <math>f(x)=\tfrac{1}{x^2}</math> with <math>f(0)=\infty</math>, is convex on the interval <math>(0, \infty)</math> and convex on the interval <math>(-\infty, 0)</math>, but not convex on the interval <math>(-\infty, \infty)</math>, because of the singularity at <math>x = 0.</math>

===Functions of ''n'' variables===
* [[LogSumExp]] function, also called softmax function, is a convex function. 
*The function <math>-\log\det(X)</math> on the domain of [[Positive-definite matrix|positive-definite matrices]] is convex.<ref name="boyd" />{{rp|74}}
* Every real-valued [[linear transformation]] is convex but not strictly convex, since if <math>f</math> is linear, then <math>f(a + b) = f(a) + f(b)</math>. This statement also holds if we replace "convex" by "concave".
* Every real-valued [[affine function]], that is, each function of the form <math>f(x) = a^T x + b,</math> is simultaneously convex and concave.
* Every [[norm (mathematics)|norm]] is a convex function, by the [[triangle inequality]] and [[Homogeneous function#Positive homogeneity|positive homogeneity]].
* The [[spectral radius]] of a [[nonnegative matrix]] is a convex function of its diagonal elements.<ref>Cohen, J.E., 1981. [https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601750-2/S0002-9939-1981-0601750-2.pdf Convexity of the dominant eigenvalue of an essentially nonnegative matrix]. Proceedings of the American Mathematical Society, 81(4), pp.657-658.</ref>