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.