Convex function: Difference between revisions

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Examples: this page inverts strong convexity with strict convexity
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==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 (I think this page inverts strong and strict convexity, pay attention!) 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&nbsp;''x''&nbsp;=&nbsp;0. It is not strictly convex.