\(A=\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\)
\(\Rightarrow A\ge\frac{1}{2}\left(x+\frac{1}{x}+y+\frac{1}{y}\right)^2\)
\(\Rightarrow A\ge\frac{1}{2}\left(x+y+\frac{1}{x}+\frac{1}{y}\right)^2\)
\(\Rightarrow A\ge\frac{1}{2}\left(x+y+\frac{4}{x+y}\right)^2\)
\(\Rightarrow A\ge\frac{1}{2}\left[4\left(x+y\right)+\frac{4}{x+y}-3\right]^2\)
\(\Rightarrow A\ge\frac{1}{2}\left[2\sqrt{4\left(x+y\right).\frac{4}{x+y}-3}\right]^2\)
\(\Rightarrow A\ge\frac{1}{2}.5^2\)
\(\Rightarrow A\ge\frac{25}{2}\)
\(Min_A=\frac{25}{2}\)
Dấu " = " xảy ra \(\Leftrightarrow x=y=\frac{1}{2}\)
