\(4x^2+2y^2+2z^2-4xy-4xz+2yz-6y-10z+34=0\\ \Leftrightarrow\left(4x^2+y^2+z^2-4xy-4xz+2yz\right)+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0\\ \Leftrightarrow\left(2x-y-z\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}2x-y-z=0\\y=3\\z=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x-8=0\\y=3\\z=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=4\\y=3\\z=5\end{matrix}\right.\)
Thay vào M
\(M=\left(4-4\right)^{22}+\left(3-4\right)^6+\left(5-4\right)^{2013}\\ M=0+1+1=2\)
\(4x^2+2y^2+2z^2-4xy-4xz+2yz-6y-10z+34=0\)
\(\Leftrightarrow\left[4x^2-4x\left(y+z\right)+\left(y+z\right)^2\right]+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0\)
\(\Leftrightarrow\left(2x-y-z\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0\)
Do \(\left(2x-y-z\right)^2,\left(y-3\right)^2,\left(z-5\right)^2\ge0\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x-y-z=0\\y-3=0\\z-5=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=4\\y=3\\z=5\end{matrix}\right.\)
\(M=\left(x-4\right)^{22}+\left(y-4\right)^6+\left(z-4\right)^{2013}=\left(4-4\right)^{22}+\left(3-4\right)^6+\left(5-4\right)^{2013}=0^{22}+\left(-1\right)^6+1^{2013}=0+1+1=2\)
