$x$ 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?
$f$ 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.