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$x$ is a vector that $x=(x_1,\ldots,x_n)$

$f(x) = -\sum_{i=1}^n \log(x_i)$

I know this is equivalent of proving that $\Pi x_{i}$ is a convex set or not, but how to prove this step? Since it seems that I can't write derivative from this, or there are other ways that I can escape from proving the convexity of the function I wrote above?

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  • $\begingroup$ Is $x$ a vector? Is $x=(x_1,\ldots,x_n)$? $\endgroup$ – ajotatxe Sep 7 '16 at 23:01
  • $\begingroup$ It is, thank you and I've updated my question. $\endgroup$ – good2know Sep 7 '16 at 23:03
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$f$ is defined on $\{x: x_i > 0\}$ and is of class $C^{\infty}$ there. Now the Hessian matrix: $$H_f(x) = \text{diag}\left( \frac1{x_1^2}, \ldots, \frac1{x_n^2}\right)$$

is clearly positive semi-definite everywhere. So the function is convex.

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