\(E=5x^2+y^2-4xy+4x-8y+1\)
\(=\left(y^2-4xy-8y\right)+5x^2+4x+1\)
\(=\left[y^2-2y\left(2x+4\right)+\left(2x+4\right)^2\right]+5x^2+4x+1-\left(2x+4\right)^2\)\(=\left(y-2x-4\right)^2+5x^2+4x+1-4x^2-16x-16\)\(=\left(y-2x-4\right)^2+\left(x^2-12x+36\right)-51\)
\(=\left(y-2x-4\right)^2+\left(x-6\right)^2-51\)
Với mọi giá trị của x ta có:
\(\left(y-2x-4\right)^2\ge0;\left(x-6\right)^2\ge0\)
\(\Rightarrow\left(y-2x-4\right)^2+\left(x-6\right)^2-51\ge-51\)
Vậy Min E = -51
Để E = 51 thì \(\left\{{}\begin{matrix}y-2x-4=0\\x-6=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y-12-4=0\\x=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=16\\x=6\end{matrix}\right.\)
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