Question

Find two numbers with a difference of 10 and a product that is a minimum. Find the minimum.

Answer 1

Answer: {-5, 5}. Product is -25, which is the minimum.

Explanation:

let a, b denote the two numbers. We know that b-a=10.

We are looking for a minimum over the product a*b.

$ab=a(a+10)= a^2+10a$

One can minimize this using derivatives. In case you have not yet had derivatives, you can also use the vertex of a parabola (since the above is a quadratic form):

$a^2+10a= (a+5)^2-25\implies x_{min}=-5$

The minimum is at the vertex a=-5 and so b=5

Their distance is 10, and their product attains the minimum value of all possiblities -25.