What is the greatest product of two numbers with sum 82

Expert Answers
embizze eNotes educator| Certified Educator

Find the greatest product of two numbers whose sum is 82:

(1) If you know calculus:

Let the numbers be x and 82-x. Then the product P=x(82-x). To maximize P we take the first derivative -- any extrema for P occur at critical points which for this function will be where the first derivative is zero.



`P'=0 ==> 2x=82==>x=41`

By the second derivative test this is a maximum so the maximum product occurs when the numbers are both 41; the product is 1681

(2) If you do not have calculus:

proceed as above to get P=x(82-x) or `P=-x^2+82x`

The graph of this function is a parabola opening down. The maximum will occur at the vertex. We can locate the x-coordinate of the vertex using `x=-b/(2a)` so `x=(-82)/(-2)=41` . (Or you can complete the square to put in vertex form.)

So x=41 and 82-x=41.

The two numbers are 41 and the product is 1681

justaguide eNotes educator| Certified Educator

The maximum product of two numbers with sum 82 has to be determined.

Let one of the numbers be x, the other is 82 - x. The product of the two numbers is x*(82 - x) = 82x - x^2. To maximize 82x - x^2, solve (82x - x^2)' = 0

=> 82 - 2x = 0

=> x = 41

The maximum product of two numbers that add up to 82 is 1681