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I have a twice differentiable function $H(x)$ for which I have already proven that:

  • $\exists !x^*:H'(x^*)=0$ and furthermore that $H''(x^*)>0$ (the only extremal is a minima). $H''(x)$ is continuous (see question asked in the comments).

Does that suffice to show that $H(x)$ is convex?

I know one would typically show that $H''(x)\geq 0$ but in this case the expression of $H''(x)$ is terribly complicated except at the point (it is unique) for which $H'(x^*)=0$.

Edit:

I can also show that $H(x)$ is monotone decreasing before $x^*$ and monotone increasing after $x^*$...

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user42397
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    You can only say that the function is convex in a neigborough of $x^*$. See the function $y=-e^{-^2}$ as an example. – Emilio Novati Jun 13 '15 at 15:42
  • @EmilioNovati I don't even see that the function has to be convex in a neighborhood of $x^*$. Couldn't you start with something convex, like $x\mapsto x^2$ and then add little wiggles that ruin the convexity but not the other properties in the question? (Of course, if the second derivative $H''$ were continuous, then it would be positive in a neighborhood and $H$ would be convex in that neighborhood, but the question assumed only the existence, not the continuity, of $H''$.) – Andreas Blass Jun 13 '15 at 16:44
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    Sorry, I don't reed well the question . I agree that we need continuity of $H''$. ....And my suggested function was obviously $ y=-e^{-x^2}$. – Emilio Novati Jun 13 '15 at 16:48
  • Ok, H'' is continuous in this case. – user42397 Jun 13 '15 at 23:05

1 Answers1

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No, it is not enough. Consider for example the function $$H(x) = \frac{e^{x^2}}{1+x^4}\in C^\infty(\mathbb{R}).$$ We have $$H'(x) = \frac{2e^{x^2}x(x^2-1)^2}{(1+x^4)^2}$$ which is zero only at $x=0$ and $H''(x)$ is positive at $x=0$ but negative at other points, so that it is not convex everywhere (to see this, simply plot it). Moreover, we have $\lim_{x\to\pm\infty}H(x)=\infty$.

Addendum: In the question, the fact that the function is monotone decreasing before $x^*$ and monotone increasing after it is a trivial consequence of the fact that the function is continuously differentiable and has a unique minimum.

Daniel Robert-Nicoud
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