Math problem.The mean of 2 numbers is 12. Find the numbers if the product is maximum.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Let the two numbers be A and B. The mean of the numbers is 12.

=> A + B = 24

=> A = (24 - B)

The product of the numbers is

P = A*B = B*(B - 24)

=> P = B^2 - 24B

To maximise the product differentiate P with respect to B

P' = 2B - 24

P' = 0

=> 2B - 24 = 0

=> B = 12

A = 24 - B = 12

For the product of the numbers to be maximum they are both equal to 12.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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