\(x+y\ge2\sqrt{xy}\Rightarrow2\sqrt{xy}\le1\Rightarrow xy\le\dfrac{1}{4}\Rightarrow\dfrac{1}{xy}\ge4\)
\(A=x^2+y^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}+4\ge\dfrac{\left(x+y\right)^2}{2}+\dfrac{2}{xy}+4\)
\(\Rightarrow A\ge\dfrac{1^2}{2}+2.4+4=\dfrac{25}{2}\)
\(\Rightarrow A_{min}=\dfrac{25}{2}\) khi \(x=y=\dfrac{1}{2}\)