Find two numbers whose difference is 104 and whose product is a minimum.

Assume two numbers to be 52 and - 52, which satisfy the difference requirement.

Now, the general numbers would be x+52, x-52, where x is a number to be determined such that the product is a minimum.

Define the product

f (x) = (x+52) (x-52)

and the derivative

f' (x) = (x^2-52^2) '=2x

For f (x) to be a maximum or minimum, f' (x) = 0 = >x=0

f" (x) = f' (2x) = 2 >0 = > f (x) is a minimum where x=0

Conclusion:

the two numbers 52 and - 52 gives a difference of 104 and their product is a minimum.