Question

Find two non-negative numbers whose sum is 53 and such that the product of the two numbers is as large as possible. What is the global maximum?

Answer 1

So we have 2 unknown numbers x and y which add up to 53,
\[\Large x+y=53\]
And they also have some product,\[\Large P=x\cdot y\]

Answer 2

For this problem, we want to `maximize` the product.
So we'll need to find `critical points` of the product function P.

So we need \(\Large P'\)
But before we go looking for the derivative, we want P in terms of one variable.

Answer 3

We can use the x+y=53 to replace one of the variables in our P function.

Answer 4

Understand what I mean by that? :o

Answer 5

okaaay so far i think i understand. so if we choose to find x first then x= 53 - y

Answer 6

good good.
So what does our product function look like now?

Answer 7

P = (53-y)(y)

Answer 8

good good good.

Answer 9

so 53y - y^2

Answer 10

Ok cool.
Now we want to take the derivative of our Product function with respect to y.

Answer 11

i understand now! hehe it was a lot easier than i thought. thank you!

Answer 12

yay \c:/