Question

Find two real numbers whose sum is 4 and whose product is a maximum.

Answer 1

so maximize xz where x + z = 4....{ x = z = 2 }.. f(x) = x [ 4 -x] is a parabola...max easy to find

Answer 2

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Sum: x + y = 4
Product = xy
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y = 4-x
Substitute that into "Product" to get:
P = x(4-x)
P = 4x-x^2
That is a quadratic with a = -1, b = 4
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Comment: That "Product" does not have a minimum; it does have a maximum.
Max occurs at x = -b/2a = -4/(2*-1) = 2
Maximum product is P = 2*(4-2)= 2*2 = 4

Answer 3

You can solve this easily without any algebra... both numbers must be positive for them to add up to 4 *and* have a maximum product... (if one is negative and the other positive, the product is negative)

Numbers that add to 4:

0 and 4
1 and 3
2 and 2

and those are your only options. Find the two that have the greatest product.