Question

tìm GTNN 

4x²+2y²+14z²-4xy+12xz-10yz-2z+6

Answer 1

\(A=4x^2+2y^2+14z^2-4xy+12xz-10yz-2z+6=\left(4x^2+y^2+9z^2-4xy-6yz+12xz\right)+\left(y^2-4yz+4z^2\right)+\left(z^2-2z+1\right)+5=\left(2x-y+z\right)^2+\left(y-2z\right)^2+\left(z-1\right)^2+5\ge5\)

\(\Rightarrow MinA=5\Leftrightarrow\hept{\begin{cases}2x-y+z=0\\y-2z=0\\z-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-\frac{1}{2}\\y=2\\z=1\end{cases}}\)