Ta có \(\frac{y}{x\sqrt{y^2+1}}=\frac{y\sqrt{xz}}{x\sqrt{y\left(x+y+z\right)+xz}}=\frac{yz}{\sqrt{x\left(y+z\right).z\left(x+y\right)}}\ge\frac{2yz}{2xz+xy+yz}\)
Đặt \(a=xy,b=yz,c=xz\)=> \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
Khi đó
\(P\ge\frac{2b}{2c+a+b}+\frac{2c}{2a+b+c}+\frac{2a}{2b+a+c}\ge\frac{2\left(a+b+c\right)^2}{b^2+c^2+a^2+3\left(ab+bc+ac\right)}\)
Xét \(P\ge\frac{3}{2}\)
=> \(4\left(a+b+c\right)^2\ge3\left(a^2+b^2+c^2\right)+9\left(ab+bc+ac\right)\)
<=> \(a^2+b^2+c^2\ge\left(ab+bc+ac\right)\)(luôn đúng )
Vậy \(MinP=\frac{3}{2}\)khi a=b=c=3=> \(x=y=z=\sqrt{3}\)