The product of two numbers is 1296. What is the maximum sum of the numbers ?

Let the numbers be x and
y.

Given that the product of the of the numbers is
1296.

x*y = 1296

==> y=
1296/ x

Let the sum of the numbers be f(x) = x+
y.

We will wrtie f(x) as a function of
x.

==> f(x) = x + (
1296/x)

==> f(x) = (x^2 +
1296)/x

Now we need to find the extreme value for
f(x).

Let us find the first
derivative.

==> f'(x) = ( 2x*x - x^2+
1296))/(x^2)

                 = ( 2x^2 - x^2 + 1296) /
x^2

                  = ( x^2 -
1296)/x^2

==> f'(x) = (x^2 -
1296)/x^2

==> (x^2 - 1296 =
0

==> x^2 =
1296

==> x= 36

Then the
function has extreme value at x= 36.

==> y=
36.

Then, the maximum sum of the two numbers
is 36+ 36 = 72.