The sum of two numbers is 36 what are the numbers if the product is a maximum.

Since the sum of the two numbers is 36, we cal assume the
numbers to be x and 36-x.

Let the product p(x) , of these
two numbers x and 36-x.

Then p(x) = x(36-x) =
36x-x^2.

P(x) = 36x=x^2 is maximum , for x = c, for which
p'(c) = 0 and P"(c) > 0.

So we differentiate P(x) 
and set P'(x) = 0 and find the the solution of P'(x) = 0. And examine if P"(c) is
< 0.

P'(x) = (36x-x^2)' = 36
-2x

P'(x) = 0 gives :  36 = 2x. Or x = c = 36/2 =
12.

P"(x) = (-2x) = -2 < 0 for all
x.

Therefore P(x) = 36x-x^2 when x=
18.

P(x) = 18(36-18) = 18^2 = 324 is the maxumum
product.