Đặt \(P=x+y+\frac{1}{x}+\frac{1}{y}\)
\(=x+y+\frac{1}{4x}+\frac{3}{4x}+\frac{1}{4y}+\frac{3}{4y}\)
\(=\left(x+\frac{1}{4x}\right)+\left(y+\frac{1}{4y}\right)+\left(\frac{3}{4x}+\frac{3}{4y}\right)\)
Áp dụng bđt AM-GM cho 2 số thực dương x,y ta được:
\(x+\frac{1}{4x}\ge2\sqrt{x.\frac{1}{4x}}=1\left(1\right)\)
\(y+\frac{1}{4y}\ge2\sqrt{y.\frac{1}{4y}}=1\left(2\right)\)
\(\frac{3}{4x}+\frac{3}{4y}\ge2\sqrt{\frac{3}{4x}.\frac{3}{4y}}=\frac{3}{2\sqrt{xy}}\left(3\right)\)
Áp dụng bđt AM-GM ta có:
\(\sqrt{xy}\le\frac{x+y}{2}=\frac{1}{2}\left(4\right)\)
Thay (4) vào (3) ta có \(\frac{3}{4x}+\frac{3}{4y}\ge3\left(5\right)\)
(1)+(2)+(5) ta được: \(P\ge3\)
Dấu"="Xảy ra \(\Leftrightarrow x=y=\frac{1}{2}\)
