\(P=\frac{1}{16}\sum\left(x-\frac{xy^3}{y^3+16}\right)=\frac{1}{16}\sum\left(x-\frac{xy^3}{y^3+8+8}\right)\ge\frac{1}{16}\sum\left(x-\frac{xy^3}{12y}\right)=\frac{1}{16}\sum\left(x-\frac{xy^2}{12}\right)\)
\(\Rightarrow P\ge\frac{1}{16}\left(3-\frac{xy^2+yz^2+zx^2}{12}\right)\)
Không mất tính tổng quát, giả sử \(x=mid\left\{x;y;z\right\}\)
\(\Rightarrow\left(x-y\right)\left(x-z\right)\le0\Leftrightarrow x^2+yz\le xy+xz\)
\(\Leftrightarrow x^2z+yz^2\le xyz+xz^2\Rightarrow xy^2+yz^2+zx^2\le xy^2+xz^2+xyz\le xy^2+xz^2+2xyz\)
\(\Rightarrow xy^2+yz^2+zx^2\le x\left(y+z\right)^2=\frac{1}{2}.2x\left(y+z\right)\left(y+z\right)\le\frac{1}{54}\left(2x+2y+2z\right)^3\)
\(\Rightarrow xy^2+yz^2+zx^2\le4\)
\(\Rightarrow P\ge\frac{1}{16}\left(3-\frac{4}{12}\right)=\frac{1}{6}\)
Dấu "=" xảy ra khi \(\left(x;y;z\right)=\left(1;2;0\right)\) và hoán vị