Find two numbers whose sum is 38 and whose product is as large as possible.

Answer 1

First, you're trying to find the sum of two numbers which is 38. If we selected x and y as two arbitrary numbers, that would mean

x + y = 38.

Solving for one of the variables, in this case we will solve for y, we get

y = 38 - x.

Second, we are asked to find the MAXIMUM that the product of the two numbers would be. The product would be represented as follows

xy

however, we already have the value for y from above

f(x) = x(38-x).

Distribute the first x

= 38x - x²

= -x² + 38x

Complete the square

= -x² + 38x

= -(x²-38x)

= -(x² - 38x +(-19)² - (-19)²)

= -(x - 19)² + 361.

The maximum product is 361 and the two numbers who's sum is 38 are 19 and 19.

Answer 2

x(38-x)=P

dP/dx=2x-38

max when dP/dx=0

x=19 and 38-x=19