$\displaystyle \begin{array}{{>{\displaystyle}l}} Ta\ có:\ M+200=2xy-yz-zx+x^{2} +y^{2} +z^{2} =\left( x^{2} +y^{2} +2xy\right) -z( y+x) +z^{2}\\ =( x+y)^{2} -2.\frac{1}{2} z( x+y) +\frac{1}{4} z^{2} +\frac{3}{4} z^{2} =\left( x+y-\frac{1}{2} z\right)^{2} +\frac{3}{4} z^{2} \geqslant 0\\ \Rightarrow M\geqslant -200\\ Vâỵ\ GTNN\ của\ M=-200.\ Dấu\ "="\ xảy\ ra\ \Leftrightarrow ( x;y;z) =( 10;−10;0) =( −10;10;0)\\ \ \end{array}$
