The mean of 2 numbers is 12 find the numbers if the product is maximum.

let the numbers be x and
y.

Given that the mean of the numbers is
12.

Then we know that the mean = sum of the numbers/total
number.

==> (x+ y) /2 =
12

Multiply by 2.

==>
x+ y = 24

==> y= (24-
x).......(1).

Now we need to determine the numbers if the
product is a maximum.

Let f(x) be the
product.

==> f(x) =
x*y

==> f(x) =
x*(24-x)

==> f(x) = 24x -
x^2.

Now to find the maximum value, we need to find the
first derivative.

==> f'(x) =
24-2x

==> 2x =
24

==> x= 12

==>
y= 12.

Then, the numbers that has the maximum product are
12 and 12.

==> 12*12 = 144 is the maximum
product.