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

the numbers are 69 and -69.