\(C=\left(x+\dfrac{1}{y}\right)^2+\left(y+\dfrac{1}{x}\right)^2=x^2+\dfrac{1}{y^2}+\dfrac{2x}{y}+y^2+\dfrac{2y}{x}+\dfrac{1}{x^2}=x^2+y^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}+2\left(\dfrac{x}{y}+\dfrac{y}{x}\right)=4+\dfrac{x^2+y^2}{x^2y^2}+2\left(\dfrac{x}{y}+\dfrac{y}{x}\right)=4+\dfrac{4}{x^2y^2}+2\left(\dfrac{x}{y}+\dfrac{y}{x}\right)\)
Áp dụng bđt cosi cho hai số dương:
\(x^2+y^2\ge2\sqrt{x^2y^2}\Rightarrow x^2y^2\le\dfrac{\left(x^2+y^2\right)^2}{4}=\dfrac{4^2}{4}=4\)
\(\dfrac{x}{y}+\dfrac{y}{x}\ge2\sqrt{\dfrac{x}{y}.\dfrac{y}{x}}=2\)
Vậy \(C\ge4+\dfrac{4}{4}+2.2=4+1+4=9\)
Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x^2=y^2\\x^2+y^2=4\end{matrix}\right.\)\(\Leftrightarrow x^2=y^2=2\Leftrightarrow x=y=\sqrt{2}\)
Vậy GTNN của C là 9
