Áp dụng BĐT cô-si, ta có:
\(\frac{1}{\left(x+1\right)}+\frac{1}{\left(y+1\right)}+\frac{1}{\left(z+1\right)}\ge3\)
\(\Leftrightarrow\frac{1}{\left(x+1\right)}\ge1-\frac{1}{\left(y+1\right)}+1-\frac{1}{\left(z+1\right)}\)
\(\Leftrightarrow\frac{y}{\left(y+1\right)}+\frac{z}{\left(z+1\right)}\ge3\sqrt{\left(\frac{yz}{\left(y+1\right)\left(z+1\right)}\right)}\)
Ta có:
\(\frac{1}{\left(x+1\right)}\ge3\sqrt{\frac{yz}{\left(x+1\right)\left(y+1\right)}}\)(1)
\(\Leftrightarrow\frac{1}{\left(y+1\right)}\ge3\sqrt{\left(\frac{xy}{\left(x+1\right)\left(z+1\right)}\right)}\)(2)
\(\Leftrightarrow\frac{1}{\left(z+1\right)}\ge3\sqrt{\left(\frac{xy}{\left(x+1\right)\left(y+1\right)}\right)}\)(3)
Từ (1); (2) và (3), ta có:
\(\frac{1}{\left(x+1\right)}+\frac{1}{\left(y+1\right)}+\frac{1}{\left(z+1\right)}\ge8\frac{xyz}{\left(x+1\right)\left(y+1\right)\left(z+1\right)}\)
\(\Rightarrow xyz\le\frac{1}{8}.\text{ dau }=\text{xay ra khi }x=y=z=\frac{1}{2}\)
