Question

What is the minimum possible product of two numbers that differ by `8` ?

Answer 1

The minimum product is `-16` and the two numbers are `-4` and `4`

Explanation:

Let the two numbers be `x` and `x+8`.

Then their product is:

`f(x) = x(x+8)`

`color(white)(f(x)) = x^2+8x`

`color(white)(f(x)) = x^2+8x+16-16`

`color(white)(f(x)) = (x+4)^2-16`

For any Real value of `x` we will have `(x+4)^2 >= 0`

Hence `f(x)` attains its minimum value `-16` when `(x+4)^2 = 0`

That is when `x = -4`

So the minimum product is `-16` and the two numbers are `-4` and `4`