\(\frac{4x^2y^2}{\left(x^2+y^2\right)^2}+\frac{x^2}{y^2}+\frac{y^2}{x^2}\ge3\)
\(\Leftrightarrow\frac{4x^2y^2}{\left(x^2+y^2\right)^2}+\frac{x^2}{y^2}+\frac{y^2}{x^2}-3\ge0\)
\(\Leftrightarrow\frac{4x^4y^4+x^4\left(x^2+y^2\right)^2+y^4\left(x^2+y^2\right)^2-3x^2y^2\left(x^2+y^2\right)^2}{x^2y^2\left(x^2+y^2\right)^2}\)
\(\Leftrightarrow4x^4y^4+x^4\left(x^4+2x^2y^2+y^4\right)+y^4\left(x^4+2x^2y^2+y^4\right)-3x^2y^2\left(x^4+2x^2y^2+y^4\right)\ge0\)
\(\Leftrightarrow4x^4y^4+x^8+2x^6y^2+x^4y^4+2x^2y^6+y^8-3x^6y^2-6x^4y^4-3x^2y^6\ge0\)
\(\Leftrightarrow x^8+y^8-x^6y^2-x^2y^6\ge0\)
\(\Leftrightarrow x^6\left(x^2-y^2\right)-y^6\left(x^2-y^2\right)\ge0\)
\(\Leftrightarrow\left(x^2-y^2\right)^2\left(x^4+x^2y^2+y^4\right)\ge0\) ( luôn đúng )
\(\Rightarrow\frac{4x^2y^2}{\left(x^2+y^2\right)^2}+\frac{x^2}{y^2}+\frac{y^2}{x^2}\ge3\)
Dấu " = " xảy ra \(\Leftrightarrow x=y\)