\(5x^2+10yz\le5\left(x^2+y^2+z^2\right)=9x\left(y+z\right)+18yz\)
\(\Leftrightarrow5x^2\le9x\left(y+z\right)+8yz\le9x\left(y+z\right)+2\left(y+z\right)^2\)
\(\Leftrightarrow5\left(\frac{x}{y+z}\right)^2-9\left(\frac{x}{y+z}\right)-2\le0\)
\(\Leftrightarrow\left(\frac{5x}{y+z}+1\right)\left(\frac{x}{y+z}-2\right)\le0\)
\(\Leftrightarrow\frac{x}{y+z}\le2\Rightarrow x\le2y+2z\)
\(\Rightarrow x+y+z\le3\left(y+z\right)\)
\(P\le\frac{2x}{\left(y+z\right)^2}-\frac{1}{\left(x+y+z\right)^3}\le\frac{4\left(y+z\right)}{\left(y+z\right)^2}-\frac{1}{\left(3y+3z\right)^3}\)
\(P\le\frac{4}{\left(y+z\right)}-\frac{1}{27\left(y+z\right)^3}=\frac{4}{y+z}-\frac{1}{27\left(y+z\right)^3}-16+16\)
\(P\le-\frac{1}{27}\left(\frac{1}{y+z}-6\right)^2\left(\frac{1}{y+z}+12\right)+16\le16\)
\(P_{max}=16\) khi \(\left(x;y;z\right)=\left(\frac{1}{3};\frac{1}{12};\frac{1}{12}\right)\)
