Lời giải:
Vì \(x,y,z\in [0;1]\Rightarrow xy; yz,xz\geq xyz\)
\(\Rightarrow P=\frac{x}{1+yz}+\frac{y}{1+xz}+\frac{z}{xy+1}\leq \frac{x}{1+xyz}+\frac{y}{1+xyz}+\frac{z}{1+xyz}=\frac{x+y+z}{xyz+1}(*)\)
\(x,y,z\in [0;1]\Rightarrow \left\{\begin{matrix} (x-1)(y-1)\geq 0\\ (xy-1)(z-1)\geq 0\end{matrix}\right.\)
\(\Rightarrow \left\{\begin{matrix} xy+1\geq x+y\\ xyz+1\geq xy+z\end{matrix}\right.\)
\(\Rightarrow xyz+2+xy\geq x+y+z+xy\)
\(\Leftrightarrow xyz+2\geq x+y+z\)
Mà: \(xyz+2\leq 2xyz+2=2(xyz+1)\)
\(\Rightarrow x+y+z\leq 2(xyz+1)(**)\)
Từ \((*); (**)\Rightarrow P\leq \frac{2(xyz+1)}{xyz+1}=2\) (đpcm)
Dấu "=" xảy ra khi \((x,y,z)=(1,1,0)\)
