\(S=x^2+5y^2+4xy-6x-16y+2031\)
\(\Rightarrow S=x^2+4y^2+y^2+4xy-6x-12y-4y+4+1918+9\)
\(\Rightarrow S=\left(x^2+4xy+4y^2\right)-6x-12y+\left(y^2-4y+4\right)+1918+9\)
\(\Rightarrow S=\left(x+2y\right)^2-6\left(x+2y\right)+\left(y-2\right)^2+1918+9\)
\(\Rightarrow S=\left[\left(x+2y\right)^2-6\left(x+2y\right)+9\right]+\left(y-2\right)^2+1918\)
\(\Rightarrow\left[\left(x+y\right)^2-2.3\left(x+2y\right)+3^2\right]+\left(y-2\right)^2+1918\)
\(\Rightarrow\left(x+y-3\right)^2+\left(y+2\right)^2+1918\)
Vì: (x+y-3)^2+(y+2)^2 > 0
=> (x+y-3)^2+(y+2)^2+1918> 1918
Dấu "=" xảy ra khi x+y-3=0;y+2=0
Ta có: y+2=0=>y=0-2=>y=-2
Thay y=-2 vào x+y-3
x+(-2)-3=0=>x-5=0=>x=0-5=>x=-5
Vậy Smin=1918 khi x=-5;y=-2