Ta có : `( ( x^3  + y^3 ) - ( x^2 + y^2 ))/( ( x - 1 ) (y - 1 ))`

`= ( ( x^3 - x^2 ) + ( y^3 - y^2 ))/( ( x - 1 )( y - 1 ))`

`= ( x^2 ( x - 1 ) + y^2 ( y - 1 ))/((x-1)(y-1))`

`= x^2/(y-1) + y^2/(x-1)`

Áp dụng BĐT AM - GM ta được :

` x^2/(y-1) + y^2/(x-1) >= 2 \sqrt{ ( x^2 y^2 )/((y-1)(x-1))} = 2 . (xy)/( \sqrt{(x-1).1} . \sqrt{(y-1).1} ) >= 2 . (xy)/(( x -1 + 1 )/2 . ( y - 1 + 1 )/2 ) = 2 . ( 4xy )/( xy ) = 2.  4 = 8`

Dấu ''='' `<=> x = y = 2`