Question

1. Tìm GTNN của A = ( 1 + x )( 1 + \(\frac{1}{y}\) ) + ( 1 + y )( 1 + \(\frac{1}{x}\) ) , với x > 0 , y > 0 thỏa mãn x² + y² = 1

Answer 1

\(A=2+x+y+\frac{1}{x}+\frac{1}{y}+\frac{x}{y}+\frac{y}{x}\)

\(A=2+x+\frac{1}{2x}+y+\frac{1}{2y}+\frac{x}{y}+\frac{y}{x}+\frac{1}{2}\left(\frac{1}{x}+\frac{1}{y}\right)\)

\(A\ge2+2\sqrt{\frac{x}{2x}}+2\sqrt{\frac{y}{2y}}+2\sqrt{\frac{xy}{yx}}+\frac{4}{2\left(x+y\right)}=4+2\sqrt{2}+\frac{2}{x+y}\)

\(A\ge4+2\sqrt{2}+\frac{2}{\sqrt{2\left(x^2+y^2\right)}}=4+3\sqrt{2}\)

\(\Rightarrow A_{min}=4+3\sqrt{2}\) khi \(x=y=\frac{1}{\sqrt{2}}\)