A= \(\frac{1}{\left(x+y\right)\left(x^2+y^2-xy\right)+xy}+\frac{4x^2y^2+2}{xy}=\)\(\frac{1}{x^2+y^2}+4xy+\frac{2}{xy}=\frac{1}{x^2+y^2}+\frac{1}{2xy}+4xy+\frac{1}{4xy}+\frac{5}{4xy}\) (1)
\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b};a+b\ge2\sqrt{ab},\frac{1}{xy}\ge\frac{4}{\left(x+y\right)^2}\)áp dụng vào trên ta được
(1) \(\ge\frac{4}{x^2+y^2+2xy}+2\sqrt{4xy.\frac{1}{4xy}}+\frac{5}{4}.\frac{4}{\left(x+y\right)^2}=4+2+\frac{5}{4}.4=11.\)
dấu '=" khi x=y = 1/2