According to law of probability or set theory :
P(A U B) = P(A) + P(B) - P(A ∩ B)
==> 1 = x + y [ As A and B are mutually exclusive so P(A ∩ B) = 0 and A U B = S so P(A U B) = 1 ]
==> y = 1 - x
So P(A).P(B) = x.y
= x.(1 - x)
= x - x^{2 }= z(say)
Now for P(A).P(B) to be maximum ,
dz / dz = 0
==> 1 - 2x = 0
==> x = 1/2
Now taking 2nd derivative we get -2 which is < 0 and hence x = 1/2 is a point of maxima..
Therefore [P(A).P(B)]_{max } = (1/2)*(1/2)
= 1 / 4