\(x^2+x^2+y^2+\frac{1}{x^2}\ge4\sqrt[4]{x^2y^2}\)
\(\Rightarrow4\sqrt[4]{x^2y^2}\le4\Rightarrow\sqrt[4]{x^2y^2}\le1\Rightarrow x^2y^2\le1\)
\(\Rightarrow-1\le xy\le1\)
\(P_{max}=1\) khi \(\left(x;y\right)=\left(1;1\right);\left(-1;-1\right)\)
\(P_{min}=-1\) khi \(\left(x;y\right)=\left(1;-1\right);\left(-1;1\right)\)