\(P=x^2+xy+y^2-2x-3y+2015\)
\(4P=4x^2+4xy+4y^2-8x-12y+8060\)
\(4P=\left(4x^2+4xy+y^2\right)+3y^2-4\left(2x+y\right)-8y+8060\)
\(4P=\left(2x+y\right)^2-4\left(2x+y\right)+4+\left(3y^2-8y+\dfrac{16}{3}\right)+\dfrac{24152}{3}\)
\(4P=\left(2x+y-2\right)^2+3\left(y-\dfrac{4}{3}\right)^2+\dfrac{24152}{3}\ge\dfrac{24152}{3}\)
\(\Rightarrow P\ge\dfrac{6038}{3}\)
\(P_{min}=\dfrac{6038}{3}\) khi \(\left\{{}\begin{matrix}2x+y-2=0\\y-\dfrac{4}{3}=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{4}{3}\end{matrix}\right.\)
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