Bài này là tìm GTLN của xyz đúng không?. Làm vậy nhé:
Ta có: \(\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\ge2\)
\(\Rightarrow\frac{1}{x+1}\ge1-\frac{1}{y+1}+1-\frac{1}{z+1}=\frac{y}{y+1}+\frac{z}{z+1}\ge2\sqrt{\frac{yz}{\left(y+1\right)\left(z+1\right)}}\left(1\right)\)
Tương tự ta có: \(\hept{\begin{cases}\frac{1}{y+1}\ge2\sqrt{\frac{zx}{\left(z+1\right)\left(x+1\right)}}\left(2\right)\\\frac{1}{z+1}\ge2\sqrt{\frac{xy}{\left(x+1\right)\left(y+1\right)}}\left(3\right)\end{cases}}\)
Nhân (1), (2), (3) vế theo vế ta được:
\(\frac{1}{\left(x+1\right)\left(y+1\right)\left(z+1\right)}\ge\frac{8xyz}{\left(x+1\right)\left(y+1\right)\left(z+1\right)}\)
\(\Leftrightarrow xyz\le\frac{1}{8}\)
Vậy GTLN là \(xyz=\frac{1}{8}\)khi \(x=y=z=\frac{1}{2}\)