Given that $b_1+b_2+\dots+b_n = 1$, how do I find the minimum value of $\frac{x_1+x_2+\dots+x_n}{x_1^{b_1}x_2^{b_2}\dots x_n^{b_n}}?$
For $n=2$ I used calculus and found the answer to be $\frac{1}{b_1^{b_1}b_2^{b_2}}.$ Extending the concept to higher values of $n$, the desired answer may be guessed as $\prod\limits_{i=1}^n \left(\frac{1}{b_i}\right)^{b_i}.$
Is there a better approach?