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

Expert Answers

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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.

==> P' = 36 - 2x

==> 36- 2x = 0

==> 2x= 36

==> x= 18

==> y= 36-18 =18

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

\p = 18*18 = 324

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