Find two positive real numbers who's product is a maximum and whose sum is 156.

Let the two numbers be x and y. The sum of the numbers is 156 so we have
  x + y = 156 (1)
 From (1) x = 156 - y
 Also let their product be A
  So A = xy
  Then A = y (156 - y) = 156y - y^2
  For A to beu a maximum then y = -b/2a. Comparing coefficients we find that a = -1 and b = 156
 So y = -156/(2* -1) = 156/2 = 78
 From (1) x + y = 156; x = 156 - y.
   x = 156 - 78 = 78. So x = 78 and y = 78