\(P=\sqrt{\dfrac{x^3}{y+3}}+\sqrt{\dfrac{y^3}{z+3}}+\sqrt{\dfrac{z^3}{x+3}}\)
\(=\dfrac{x^2}{\sqrt{x\left(y+3\right)}}+\dfrac{y^2}{\sqrt{y\left(z+3\right)}}+\dfrac{z^2}{\sqrt{z\left(x+3\right)}}\)
\(\ge\dfrac{\left(x+y+z\right)^2}{\sqrt{x\left(y+3\right)}+\sqrt{y\left(z+3\right)}+\sqrt{z\left(x+3\right)}}\)
Xét:
\(\left(\sqrt{x\left(y+3\right)}+\sqrt{y\left(z+3\right)}+\sqrt{z\left(x+3\right)}\right)^2\le\left(1^2+1^2+1^2\right)\left(xy+3x+yz+3y+xz+3z\right)\)
\(=3\left(9+xy+yz+xz\right)\)
\(=27+3\left(xy+yz+xz\right)\le27+\left(x+y+z\right)^2=36\)
\(\Rightarrow\sqrt{x\left(y+3\right)}+\sqrt{y\left(z+3\right)}+\sqrt{z\left(x+3\right)}\le6\)
\(P\ge\dfrac{3}{2}\)
\("="\Leftrightarrow x=y=z=1\)
