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

*print*Print*list*Cite

The sum of two numbers is 72. If one the numbers is x, the other number is 72 - x. The product of the two numbers is y = x*(72 - x)

y = 72x - x^2

To maximize y, find the derivative y' and solve y' = 0.

y' = 72 - 2x

72 - 2x = 0

x = 36

The second derivative of y is always negative indicating that a maximum value has been found.

The maximum product of the numbers is 36*36 = 1296

Let's note the numbers as x and y:

x+ y = 72 (1)

We'll isolate y to the left side and we'll get:

y = 72 - x

We'll write their product:

x*y = x(72 - x)

We'll note the expression of the product:

f(x) = x(72 - x)

We'll remove the brackets:

f(x) = -x^2 + 72x

Now, the value of the product is maximum (because the coefficient of x^2 is negative) if the derivative of f(x) is cancelling (equals zero).

We'll calculate f'(x):

f'(x) = -2x + 72

We'll put f'(x) as 0:

-2x + 72 = 0

We'll divide by -2:

x - 36 = 0

x = 36

The other number is y = 72 - x.

y = 72 - 36

y = 36

The numbers x and y are equal.

f(x) = x*y

f(36) = 36*36

f(36) = 1296