Question

Find two numbers whose sum is 14 and whose product is the maximum possible value.

What two numbers yield this product?

______.

Answer 1

Let x = one of the numbers. Then (14-x) is the other one.

So, we have to maximize y = x(14-x) = -x^2+14x

The graph of y = -x^2+14x is a parabola opening downward with x-intercepts (0,0) and (14,0).

The x-coordinate of the maximum point lies halfway between 0 and 14.

So, the two numbers are 7 and 7.   

ALTERNATE METHOD

Lets make a chart of x and y in which the sum is 14, and shows their product.

x | y | x*y

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0 14 0

1 13 13

2 12 24

3 11 33

4 10 40

5 9 45

6 8 48

7 7 49

Based on this chart, look for the maximum product. Then look for the x and y values that give that product.

Hence, the two numbers are 7 and 7 .