\(D=2x^2+4y^2+4xy+2x+4y+9=x^2+4xy+4y^2+2x+4y+1+x^2+8=\left(x+2y\right)^2+2\left(x+2y\right)+1+x^2+8=\left(x+2y+1\right)^2+x^2+8\)
Do : \(\left\{{}\begin{matrix}\left(x+2y+1\right)^2\ge0\forall xy\\x^2\ge0\forall x\end{matrix}\right.\)\(\Rightarrow\left(x+2y+1\right)^2+x^2\ge0\)
\(\Leftrightarrow\left(x+2y+1\right)^2+x^2+8\ge8\)
\(\Rightarrow D_{Min}=8."="\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=-\dfrac{1}{2}\end{matrix}\right.\)
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