Let the larger number of two numbers be x. Then the other number is `(x-5)`
The product of the two numbers is `y = x(x-5)` .
We should find x such that y is a minimum. To find that we can use differentiation. Let's first find the first derivative and find the extremee points of y and check for the sign of second derivative to check for the minimum.
`y = x(x-5)`
`y = x^2-5x`
`(dy)/(dx) = 2x-5`
For maxima and minima `(dy)/(dx) = 0`
`2x-5 = 0`
`x = 5/2`
`(d^2y)/(dx^2) = 2`
This is always positive. Therefore at `x = 5/2` .
Therefore the two numbers are `2.5` and `(5-2.5)` which is `-2.5` .
The answer is 2.5 and -2.5. The lowest product is -6.25
-4 & 1
The difference between them is 5 and from what I have worked out it is the pair of numbers with the smallest possible product.