Find two integers whose sum is 12 and whose product is maximum.

We'll note the integers as x and
y.

The sum of the integers is
12.

x + y = 12

y = 12 -
x

We also know that the product of integers is a
maximum.

We'll write the product of integers
as:

P = x*y

We'll substitute y
by (12-x) and we'll create the function p(x):

p(x) =
x*(12-x)

We'll remove the brackets and we'll
get:

p(x) = 12x - x^2

The
function p(x) is a maximum when x is critical, that means that p'(x) =
0

We'll calculate the first derivative for
p(x):

p'(x) = (12x -
x^2)'

p'(x) = 12 - 2x

p'(x) =
0

12 - 2x = 0

We'll divide by
2:

6 - x = 0

We'll subtract 6
both sides:

-x = -6

We'll
divide by -1:

x =
6

So, x is the critical value and the
integers are x = 6 and y = 6.