Find two integers whose difference is 12 and whose product is a minimum.

The difference of the two integers is 12. Let one of them
be A, the other is 12 + A.

We have to minimize A*(12 +
A)

The value of A for which P = A^2 + 12A is the minimum
can be found by solving P' = 0 for A and ensuring that P''(A) is
positive.

P' = 2A + 12

2A + 12
= 0

=> A = -6

P'' =
2

At A = -6, P'' is positive, this indicates the minimum
value is at A = -6

The integers are A = -6 and A + 12 =
6

The required integers are -6 and
6