Originally Posted by

**quantoembryo** Hello all, I got another maximization problem I would like to go over.

Let 0<a<b. Find points (x, y, z) where a<x<y<z<b that maximize

$\displaystyle u=\frac{xyz}{(a+x)(x+y)(y+z)(z+b)}$

First, I said since a>0, and a is the smallest in the domain, let

$\displaystyle v=ln(u)$

and the maximum should be the same.

Hence,

$\displaystyle v(x, y, z, a, b)=lnx+lny+lnz-ln(a+x)-ln(x+y)-ln(y+z)-ln(z+b)$

where

$\displaystyle v_1(x, y, z, a, b)=\frac{1}{x}-\frac{1}{a+x}-\frac{1}{x+y}$

$\displaystyle v_2(x, y, z, a, b)=\frac{1}{y}-\frac{1}{x+y}-\frac{1}{y+z}$

$\displaystyle v_3(x, y, z, a, b)=\frac{1}{z}-\frac{1}{y+z}-\frac{1}{z+b}$

$\displaystyle v_4(x, y, z, a, b)=\frac{-1}{a+x}$

$\displaystyle v_5(x, y, z, a, b)=\frac{-1}{z+b}$

Does this seem reasonable thus far?