\(3-S=1-\frac{x}{x+1}+1-\frac{y}{y+1}+1-\frac{z}{z+1}=\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\)
\(\Rightarrow3-S\ge\frac{9}{x+y+z+3}=\frac{9}{4}\)
\(\Rightarrow S\le3-\frac{9}{4}=\frac{3}{4}\)
\(\Rightarrow S_{max}=\frac{3}{4}\) khi \(x=y=z=\frac{1}{3}\)