Let P = xy then the question becomes, “What is the maximum value of P?”

If x + y = a then x = a - y so

P = (a - y) * y

P = ay - y^2

P’ = a - 2y

The maximum occurs when P’ = 0 or

0 = a - 2y

2y = a

y = a / 2

If y = a/2 then x = a - a / 2 = a/2.

Therefore, the maximum occurs when x = y = a / 2.