Question

Cho x, y, z>0 thỏa mãn xy+yz+zx=5. Tìm GTNN của P=\(\frac{3x+3y+2z}{\sqrt{6\left(x^2+5\right)}+\sqrt{6\left(y^2+5\right)}+\sqrt{z^2+5}}\)

Answer 1

\(x^2+5=x^2+xy+yz+zx=\left(x+y\right)\left(x+z\right)\)

\(\Rightarrow P=\frac{3x+3y+2z}{\sqrt{6\left(x+y\right)\left(x+z\right)}+\sqrt{6\left(x+y\right)\left(y+z\right)}+\sqrt{\left(x+z\right)\left(y+z\right)}}\)

\(P=\frac{3x+3y+2z}{\sqrt{\left(3x+3y\right)\left(2x+2z\right)}+\sqrt{\left(3x+3y\right)\left(2y+2z\right)}+\sqrt{\left(x+z\right)\left(y+z\right)}}\)

\(P\ge\frac{2\left(3x+3y+2z\right)}{3x+3y+2x+2z+3x+3y+2y+2z+x+z+y+z}\)

\(P\ge\frac{2\left(3x+3y+2z\right)}{9x+9y+6z}=\frac{2\left(3x+3y+2z\right)}{3\left(3x+3y+2z\right)}=\frac{2}{3}\)

\(P_{min}=\frac{2}{3}\) khi \(\left\{{}\begin{matrix}x=y=1\\z=2\end{matrix}\right.\)