Question

The product of two numbers is 48. Find the sum of the two numbers, S(x), as a function of one of the numbers, x

S(x) =

Find the positive numbers that minimize the sum and list them in increasing order.

Answer 1

I thought it would be x + 48/x

But that only leaves 1 positive value for x when I take derivative and set to 0.

Answer 2

One of the answers is sqrt 48

Can't figure out the other.

Answer 3

who said there must be two values where f ' (x) = 0?

Answer 4

The online system my teacher is using, webassign
There are two blank answers

Answer 5

And it says they both have to be positive. -sqrt48 gave an incorrect answer.

Answer 6

If one number is x, the other number is 48/x.

If one number is -x, then the other number is -48/x.

So one pair can be both being positive. The second pair is where both numbers are negative.

Answer 7

So how do I figure out the positive numbers?

Answer 8

I've taken a screenshot of the problem

Answer 9

You want to minimize the sum...that means you want to find the minimum value for x + (48/x).

And you'll need to take the derivative of x + (48/x) and find critical points, etc.

Answer 10

Don't you get for the derivative 1 - (48/x^2)?

Answer 11

Yeah. I only managed to find 2 critical points, 1 of which was negative. I need 2 positive values.

Answer 12

which is (x^2 - 48)/(x^2)

Answer 13

Yeah I did, the only numbers I can think of that can make that true are sqrt48 and -sqrt48.

Answer 14

so you 3 critical points altogether; x = 0, x = sqrt (48); and x = - sqrt(48).

Answer 15

Oh 0? Wouldn't that give an error though?

Answer 16

You want to find critical points by algebra, not by having to guess/think about it. Set the numerator equal to zero, and denominator equal to zero. Those are the critical points.

Now use the First Derivative Test to determine where there's a rel min. Thats what you are looking for.

Answer 17

0 is still a critical point..because that';s where f ' (x) is undefined.

Answer 18

A critical point is a point where f ' (x) = 0 or undefined.

Answer 19

And that would be considered a minimum? The later parts of the problem are asking what f`(x) would be at that point. How would I answer that?

Answer 20

No. to find a (rel) min/max, you need to use the First Derivative Test.

Answer 21

the critical points are the "possible candidates" for a rel min/max.

Answer 22

Yeah, I get that, and when I do that I find that 0 is undefined, so that doesn't really work for the problem, right?

Answer 23

I'm sorry if I'm asking dumb questions. I'm a bit lost.

Answer 24

Again, I did post a screenshot of the question if I'm not making it clear enough.

Answer 25

so - sqrt(48) is a min

Answer 26

Yeah, but its asking for 2 positive mins.

Answer 27

When I entered in -sqrt48 it gave me an incorrect answer

Answer 28

I just read what your screenshot said.

Answer 29

Why don't you use the First derivative Test to see if - sqrt(48) is a min or not?

Answer 30

Well it is, the problem is that it's asking to "Find the positive numbers that minimize the sum "

Answer 31

seems that - sqrt(48) will not be a min (it will be a max)

Answer 32

sqrt (48) will be a rel. min. according to the First Derivative Test.

Answer 33

Oh. Then yeah. You're right. Still need to find a different minimum

Answer 34

Hi bibby. Here's a screenshot of the problem

Answer 35

I'm so confused