a)\(P=x^2+4x+2xy+3y^2+5y+2017\)
\(=x^2+2xy+y^2+4y+4+4x+2y^2+y+\dfrac{1}{8}+\dfrac{16103}{8}\)
\(=\left(x+y+2\right)^2+2\left(y^2+\dfrac{y}{2}+\dfrac{1}{16}\right)+\dfrac{16103}{8}\)
\(=\left(x+y+2\right)^2+2\left(y+\dfrac{1}{4}\right)^2+\dfrac{16103}{8}\ge\dfrac{16103}{8}\)
Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x=-\dfrac{7}{4}\\y=-\dfrac{1}{4}\end{matrix}\right.\)
b)\(Q=-x^2+4x-3y^2+6y+2017\)
\(=-x^2+4x-4-3y^2+6y+3+2024\)
\(=-\left(x^2-4x+4\right)-\left(3y^2-6y-3\right)+2024\)
\(=-\left(x-2\right)^2-3\left(y^2-2y-1\right)+2024\)
\(=-\left(x-2\right)^2-3\left(y-1\right)^2+2024\ge2024\)
Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)
Ta có:
\(P=x^2+4x+2xy+3y^2+5y+2017\)
\(=x^2+2x\left(y+2\right)+\left(y+2\right)^2+2y^2+y+2013\)
\(=\left[x+\left(y+2\right)\right]^2+2\left(y^2+y+0,25\right)+2012,5\)
\(=\left(x+y+2\right)^2+2\left(y+0,5\right)^2+2012,5\ge2012,5\)
Dấu "=" xảy ra khi:
\(\Leftrightarrow\left\{{}\begin{matrix}x+y+2=0\\y+0,5=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=-0,5\\x=-1,5\end{matrix}\right.\)
Vậy \(minP=2012,5\) khi \(\left\{{}\begin{matrix}y=-0,5\\x=-1,5\end{matrix}\right.\)
Ta có:
\(Q=-x^2+4x-3y^2+6y+2017\)
\(=-\left(x^2-4x+4\right)-3\left(y^2-2y+1\right)+2024\)
\(=-\left(x-2\right)^2-3\left(y-1\right)^2+2024\le2024\)
Dấu "=" xảy ra khi \(\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\y-1=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)
Vậy \(maxQ=2024\) khi \(\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)