t
\(x^2+y^2+z^2=xy+3y+2z-4\)
\(\Leftrightarrow4x^2+4y^2+4z^2=4xy+12y+8z-16\)
\(\Rightarrow4x^2+4y^2+4z^2-4xy-12y-8z+16=0\)
\(\Rightarrow\left(4x^2-4xy+y^2\right)+\left(3y^2-12y+12\right)+\left(4z^2-8z+4\right)=0\)
\(\Rightarrow\left(2x-y\right)^2+3\left(y^2-4y+4\right)+4\left(z^2-2z+1\right)=0\)
\(\Rightarrow\left(2x-y\right)^2+3\left(y-2\right)^2+4\left(z-1\right)^2=0\)
Vì \(\left(2x-y\right)^2+3\left(y-2\right)^2+4\left(z-1\right)^2\ge0\)
Xảy ra khi \(\left\{\begin{matrix}2x-y=0\\y-2=0\\z-1=0\end{matrix}\right.\)\(\Rightarrow\left\{\begin{matrix}x=z=1\\y=2\end{matrix}\right.\)
Suy ra \(\left\{\begin{matrix}x_0=z_0=1\\y_0=2\end{matrix}\right.\)\(\Rightarrow x_0+y_0+z_0=1+1+2=4\)
