The product of two positive numbers is 124. If one of the numbers is x, the other number is `124/x` . The sum of the two numbers is `S = x + 124/x`

The first derivative of S is `S' = 1 - 124/x^2`

Solving S' = 0

=> `1 - 124/x^2 = 0`

=> `x = sqrt 124`

At `x = sqrt 124` , S'' is positive. This indicates that the value of S is minimum when `x = sqrt 124` . The maximum value that S can take on is infinity.

The maximum value of the sum of the two numbers is infinity.