When P(A)P(B) is maximum A,B together should complete whole universal space( P(A∪B)=1, also in the question it is mentioned that A∪B=S ,so P(A U B)=1) and P(A),P(B) not equal to 0
It is given that A,B are mutually exclusive events(P(A∪B)= P(A) + P(B) ). So, if we let P(A)=x then,
P(A∪B)= P(A) + P(B) or, 1= x +P(B) or, P(B)=1-x. Hence, P(A)P(B)= x.(1-x)
derivative of x(1-x)= 1-2x, So, when P(A)P(B) is maximum, 1-2x=0 or, x=1/2, Hence x(1-x)= 1/2 * 1/2 = 1/4=0.25