Question

[Hà]

Giúp t vs :((

Answer 1

[Nguyễn Minh Quang]

ta có :

\(S=4-2\left(x+y\right)+xy=4-2\left(x+y\right)+xy+\frac{x^2+y^2-1}{2}\)

\(=\frac{\left(x+y\right)^2}{2}-2\left(x+y\right)+\frac{7}{2}=\frac{1}{2}\left(x+y-2\right)^2+\frac{3}{2}\)

mà ta có : \(\left(x+y\right)^2\le2\left(x^2+y^2\right)=2\Rightarrow-\sqrt{2}\le x+y\le\sqrt{2}\)

\(\Rightarrow-2-\sqrt{2}\le x+y-2\le\sqrt{2}-2< 0\)

\(\Rightarrow\left(2-\sqrt{2}\right)^2\le\left(x+y-2\right)^2\le\left(\sqrt{2}+2\right)^2\)

hay \(6-4\sqrt{2}+\frac{3}{2}\le S\le6+4\sqrt{2}+\frac{3}{2}\)