\(C=x^2+5y^2-2xy+4y+3\)
\(=x^2+4y^2+y^2-2xy+4y+2+1\)
\(=\left(x^2-2xy+y^2\right)+\left(4y^2+4y+1\right)+2\)
\(=\left(x-y\right)^2+\left(2y+1\right)^2+2\)
Ta có: \(\left(x-y\right)^2\ge0\) ; \(\left(2y+1\right)^2\ge0\)
\(\Rightarrow\left(x-y\right)^2+\left(2y+1\right)^2\ge0\)
\(\left(x-y\right)^2+\left(2y+1\right)^2+2\ge2\)
Vậy GTNN của C là 2
Dấu \("="\) xảy ra khi :
\(2y+1=0\Rightarrow2y=0-1=-1\Rightarrow y=\dfrac{-1}{2}\)
hoặc \(x-y=0\)\(\Rightarrow x=y=-\dfrac{1}{2}\)
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