Ta có \(C=5x^2+y^2+z^2-4x-2xy-z-1\)
\(=x^2-2xy+y^2+4x^2-4x+1+z^2-z+\dfrac{1}{4}-1-\dfrac{1}{4}-1\)
\(=\left(x-y\right)^2+\left(2x-1\right)^2+\left(z-\dfrac{1}{2}\right)^2-\dfrac{9}{4}\)
Ta có \(\left(x-y\right)^2\ge0;\left(2x-1\right)^2\ge0;\left(z-\dfrac{1}{2}\right)^2\ge0\)
=> \(C\ge-\dfrac{9}{4}\)
=> C đạt giá trị nhỏ nhất là \(-\dfrac{9}{4}\) khi
\(\left\{{}\begin{matrix}x-y=0\\2x-1=0\\z-\dfrac{1}{2}=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=y=\dfrac{1}{2}\\x=\dfrac{1}{2}\\z=\dfrac{1}{2}\end{matrix}\right.\)
=> \(x=y=z=\dfrac{1}{2}\)
Vậy MinC = \(-\dfrac{9}{4}\)khi x=y=z = \(\dfrac{1}{2}\)
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