\(xy+2=\frac{1}{xy}+x^4+y^4\ge\frac{1}{xy}+2x^2y^2\)
\(\Leftrightarrow2x^2y^2+\frac{1}{xy}-xy-2\le0\)
\(\Leftrightarrow\frac{2\left(xy\right)^3-\left(xy\right)^2-2\left(xy\right)+1}{xy}\le0\)
\(\Leftrightarrow\frac{\left(xy-1\right)\left(xy+1\right)\left(2xy-1\right)}{xy}\le0\)
\(\Rightarrow\left[{}\begin{matrix}-1\le xy< 0\\\frac{1}{2}\le xy\le1\end{matrix}\right.\)
\(P_{min}=-1\) khi \(\left(x;y\right)=\left(-1;1\right);\left(1;-1\right)\)
\(P_{max}=1\) khi \(\left(x;y\right)=\left(1;1\right);\left(-1;-1\right)\)
