\(\left(x+y\right)xy=x^2+y^2-xy\Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{x^2}+\dfrac{1}{y^2}-\dfrac{1}{xy}=\dfrac{3}{4}\left(\dfrac{1}{x}-\dfrac{1}{y}\right)^2+\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\ge\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\Rightarrow\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}\le4\)
\(A=\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}-\dfrac{1}{xy}\right)=\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\le4^2=16\)
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