Asymptotic behavior of convex functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:12:41Z http://mathoverflow.net/feeds/question/107586 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107586/asymptotic-behavior-of-convex-functions Asymptotic behavior of convex functions Henrique 2012-09-19T16:13:07Z 2012-09-19T23:04:27Z <p>Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly positive. If $x_n$ is a sequence of points such that $f(x_n)\rightarrow 0$, show that (or give a counterexample) the gradient $\nabla f(x_n)$ also tends to zero. </p> http://mathoverflow.net/questions/107586/asymptotic-behavior-of-convex-functions/107625#107625 Answer by Alexandre Eremenko for Asymptotic behavior of convex functions Alexandre Eremenko 2012-09-19T22:57:09Z 2012-09-19T22:57:09Z <p>A counterexample is $$f=\sqrt{y^2+e^{-x}}.$$ You can verify by computing the second derivatives that this is convex. As a sequence $x_n$ you can take $(n,1/n)$. Then $f(x_n)\to 0$ but the derivative with respect to $y$ tends to 1. Thus the gradient does not tend to 0.</p>