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=82x-x^2`

`P'=82-2x`

`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

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**

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