Find two positive real numbers whose product is a maximum and whose sum is 156

let the two number be x and y
x + y = 156
y = 156 - x

Product of the 2 numbers:

x(156 - x) = 

This maximum value occurs when x = -b / (2a).  [This is a standard rule]

a = -1 and b = 156
x =  
x = 78
y = 156 - 78 = 78

product = 78 x 78 = 6084 

So the two positive numbers are 78 and 78.