Find two numbers whose difference is 138 and whose product is a minimum.

Let the numbers be x and x-138, then product

p (x) = x (x-138)

Method 1: No calculus

The minimum can be found by completing the square:

p (x) = (x-69) ^2-69^2

Therefore the numbers are 69 and 69-138=-69.

Method 2: with calculus

The minimum of p (x) is when p' (x) = 0 = >

p' (x) = (x (x-138)) '=2x-138=0 = > x=69, therefore x-138=-69

Answer: the numbers are 69 and - 69.