Question 1179956
Let the number be {{{x}}} and{{{ y}}};

Given that the sum is {{{36}}}

==> {{{x + y= 36}}}

We will write as function of {{{y}}}:

==> {{{y= 36-x}}} .............(1)

Now we need to find the numbers such that their product is a maximum.

Let {{{p}}} be the product:

==> {{{p = x*y}}}

But {{{y= (36-x)}}}

==> {{{p = x*(36-x)}}}

==> {{{p = 36x - x^2}}}

Now we need to find the maximum point of{{{ P}}}

Since the sign of {{{x^2}}} is negative, then the function has a {{{maximum}}} 

find roots:

==>{{{ 0 =  36x - x^2}}}

==> {{{ 0 =  x(36 - x)}}}

==> real solutions are {{{x=0}}} or {{{x= 18}}}

disregard {{{0}}} solution, so

==> {{{x= 18}}}

then go to

{{{y= 36-x}}} .............(1), substitute {{{x}}}

==> {{{y= 36-18 =18}}}

Then the numbers are {{{18}}} and {{{18}}} and the maximum product is:

{{{p = 18*18 = 324}}}