M=\(\left(1-\frac{1}{x^2}\right)\left(1-\frac{1}{y^2}\right)
\)
=\(\left(1-\frac{1}{x}\right)\left(1+\frac{1}{x}\right)\left(1-\frac{1}{y}\right)\left(1+\frac{1}{y}\right)\)
=\(\frac{x-1}{x}.\frac{y-1}{y}.\left(1+\frac{1}{x}\right).\left(1+\frac{1}{y}\right)\)
=\(\frac{x-\left(x+y\right)}{x}.\frac{y-\left(x+y\right)}{y}.\left(1+\frac{1}{x}\right).\left(1+\frac{1}{y}\right)\)
=\(\frac{\left(-x\right).\left(-y\right)}{xy}.\left(1+\frac{1}{x}\right).\left(1+\frac{1}{y}\right)\)
=\(\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)=1+\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}\) \(\ge1+\frac{4}{x+y}+\frac{1}{\frac{\left(x+y\right)^2}{4}}=1+\frac{1}{4}+\frac{1}{\frac{1}{4}}=9\)
Vậy GTNN của M là 9 khi x=y=\(\frac{1}{2}\)
