Two numbers have a sum of 72. What is their product if it is a maximum?

The sum of the two numbers is
72.

 So we assume x is a number and 72-x is the other
number.

Their product  P(x) =
x(72-x).

We have to find the number x such that x(72-x) is
maximum.

We know by calculus that P(x) is maximum  for x=
c,  if P'(c) = 0 or {x(72-x}' = 0 and p"(c) <
0.

P'(c) = 0 gives : {x(72-x)}' = 0. Or {72x-x^2}' = 0, or
72-2x= 0, or 72=2x.  So c = x = 72/2 = 36.

P"(x) = (72-2x)'
= -2.  So P"(c) = -2 which is < 0.

Therefore, if x=
c = 36, then substituting into the function P(x):

 x(72-x)
--> 36(72-36) = 36^2 is maximum.

So the maximum
product numbers whose sum is 72  is 36^2 = 1296 .