For positive real numbers x, y and z compute the maximum value of
\(\[\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2}.\]\)
We can write the expression as [\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2} = \frac{xy(z(x + y) + 1)}{(x + y)^2}.]By AM-GM, [xy(z(x + y) + 1) \le \sqrt{xy \cdot xy \cdot (z(x + y) + 1)^2} = \sqrt{(xy)(z(x + y))^2 (1 + 1/z)^2} = \sqrt{(x + y)^2 (y + z)^2},]so [\frac{xy(z(x + y) + 1)}{(x + y)^2} \le 1.]
Equality occurs when z=yx+y, so the maximum value is 1.
