as x + y = 1,we have 2*sqrt(xy) = 1/(1/4)^2 = 16 ----------- (2)
then we have original expression:
x^2 + 2 + 1/x^2 + y^2 + 2 + 1/y^2
= x^2 + y^2 + (x^2 + y^2)/(xy)^2 + 4
as (1) and (2),the expression above can be rewritten as
.
>= 1/2 + 16/2 + 4 = 25/2
so the required inequality is achieved and equality is valid when and only when x = y = 1/2