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.

We have the mean of two numbers as 12. We have to find the numbers if the product is maximum.

Let the numbers be a and b.

12 = (a +b)/2

=> a = 24 - b

Now the product of the two numbers is b*(24 - b) = 24b - b^2

The first derivative of 24b - b^2 = 24 - 2b

The second derivative is -2.

Therefore when we equate 24 - 2b to 0, the value of b we get will lead to the maximum value.

So 24 - 2b = 0

=> 2b = 24

=> b = 12

Therefore the two numbers are 12 and 12.

**The maximum product is achieved when both the numbers are equal and their value is 12.**

Since the mean of two numbers is twelve, the sum of the 2 numbers = 24.

If one number is x, then the other number is 24-x.

So the product of two numbers P(x) = x(24-x) = 24x-x^2.

P(x) = 24x-x^2

P(x) = -(x^2--24).

P(x) = - {(x-12)^2 +12^2} +12^2. We added and subtracted 12^2.

P(x) = 144 - (x-12)^2

P(x) = 144 is maximumum when x= 12, as -(x-12)^2 is a negative number for all x and (x-12)^2 = 0 when x= 12.

Therefore the maximum product p(x) -144 when x= 12.

P(x

The numbers are a and b.

Since you did not specified what kind of mean(arithmetical or geometric mean), we'll suppose that we have the arithmetical mean:

a + b = 2*12

a + b = 24

a = 24 - b

The product is:

P = a*b

P = (24 - b)*b

We'll remove the brackets and we'll have:

P = 24b - b^2

We'll consider the function P(b). For P(b) to have an extreme, P'(b) = 0

P'(b) = 24 - 2b

P'(b) = 0

24 - 2b = 0

-2b = -24

b = 12

a = 24 - 12

a = 12

**For the product of the numbers to be maximum, the numbers are equal:a = b = 12.**