Consider the following problem Find two numbers whose sum is

Consider the following problem: Find two numbers whose sum is 21 and whose product is as large as possible. Experiment with the problem by making a table like the one following, showing the product of different pairs of numbers that add up to 21. On the basis of the evidence in your table, estimate the answer to the problem. (Round your answers to one decimal place. Enter your answers as a comma-separated list.) Find a function that models the product in terms of x, the first of the two numbers. p(x) = Use your model to solve the problem. Compare with your answer to part (a). (Enter your answers as a comma-separated list.)

Solution

a.

when x=10 and y= 21-10=11 product is xy=110

b. x+y=21

y=21-x

p(x)=x*y=x(21-x)=21x -x²

c. The maximum value is at the vertex

And vertex is at ,x=-21/-2=10.5

And at x=10.5, product p(x)=21(10.5)-10.5²=110.25

Therefore maximum product is 110.25 at x=10.5