How can I solve this exercise?
$min \quad {\sum_{i=1}^{n}{\frac{c_i}{x_i}}} $
$s.t. \quad \sum_{i=1}^n{a_ix_i}=b $
$x_i \geq 0 $
$a_i,b_i,c_i >0$
My attempts.
I think I should calculate the derivatives: $\sum\limits_{i=1}^n\frac{\partial f}{\partial xi}=\sum_{i=1}^n\frac{c_i}{x_i^2}$, but then I don't know how to go on and how to use KKT conditions with summatories
By C-S $$\sum_{i=1}^n\frac{c_i}{x_i}=\frac{1}{b}\sum_{i=1}^na_ix_i\sum_{i=1}^n\frac{c_i}{x_i}\geq\frac{1}{b}\left(\sum_{i=1}^n\sqrt{a_ic_i}\right)^2.$$ The equality occurs for $$\left(\sqrt{a_1x_1},\sqrt{a_2x_2},...,\sqrt{a_nx_n}\right)||\left(\sqrt{\frac{c_1}{x_1}},\sqrt{\frac{c_2}{x_2}},...,\sqrt{\frac{c_n}{x_n}}\right),$$ which says that we got a minimal value.