$\underset{x}{\operatorname {argmax}} \displaystyle \sum^n_{j=1}a_j\operatorname{ln}(x_j)$ s.t. $\displaystyle \sum^n_{j=1}\frac{x_j}{b_j}=1$
This is a multivariate lagrangian homework question, the last on my problem set, that I have gotten stuck on because we didn't really go over how to do it in class.
So I have written the lagrangian: $L=\displaystyle \sum^n_{j=1} a_j\operatorname {ln}(x_j)+\lambda(\sum^n_{j=1}\frac{x_j}{b_j}-1)$
Taking the derivative with respect to $x_j$ there are $n$ equations of the form: $\displaystyle \frac{a_j}{x_j}+\frac{\lambda}{b_j}=0$ and one equation of the form $\displaystyle \sum^n_{j=1}\frac{x_j}{b_j}-1=0$
And I don't know what to do from here. I could do $\lambda=\displaystyle -\frac{a_j b_j}{x_j}$ and add them up but doesn't seem to help much because of the $a_j$ term in there. Thanks for any help on this.
I did $x_j=\displaystyle -\frac{a_jb_j}{\lambda}$ and then we get $\displaystyle \sum-\frac{a_j}{\lambda}=1$ so $\lambda=-\displaystyle \sum a_j$. Then $x_j=\displaystyle \frac{a_jb_j}{\sum a_j}$. Is this the answer?