Question

Find two real numbers whose sum is 4 and whose product is a maximum.

Answer 1

I think it's time to use your calculus.

Answer 2

I think you can do it without calculus.... if you set up an equation for the sum, then solve in terms of one of the variables.....

then you use that in the expression for the product, and you will get a quadratic. Quadratics are always max'd or min'd at the vertex.

Answer 3

well, it depends on the solution req'd

Answer 4

Well, how can i get the maximum product?

Answer 5

What is your equation for the sum?

Answer 6

0=x^2-8x+x, but i'm not pretty sure about it. Please help me

Answer 7

Whoops. This is a bit different than I was thinking. What class is this for? That will help us understand what method(s) you are supposed to be using.

Answer 8

http://www.wolframalpha.com/input/?i=Maximize [{xy%2C+x%2By%3D4}%2C+{x%2C+y}]

Answer 9

Sorry for the late reply :) uhmm, the topic is "WORD PROBLEMS INVOLVING QUADRATIC FUNCTIONS"

Answer 10

@DebbieG are you still there?

Answer 11

I am... I figured the idea here is to maximize a quadratic, but the method I was trying it not giving the right answer. I am still thinking about it, but maybe someone else has an idea?

Answer 12

@skullpatrol ..... ?

Answer 13

hmm. I see

Answer 14

I have an idea... give me a sec....

Answer 15

Okaay :) i'll wait

Answer 16

OMG it's been so long since I did this, I was just not looking at my solution correctly. Hah! what I did in the first place is fine.

You need a sum of 4, right? so let's use x and y. then we want:

x+y=4

Solve that for x: y=4-x right?

Answer 17

Now at the same time, we want to maximize the product. Let's call the product P(x,y) so it is:

P(x,y)=xy

But we have an expression for y in terms of x: y=4-x

So we can write P(x,y) as just a function of x:

P(x)=x(4-x)

Answer 18

Now P(x) is a quadratic in x. Quadratics are max'd or min'd at their VERTEX. So find the vertex of P(x).

That will give you the value of x, and the maximum of the product. E.g., the vertex is:

(x, P(x)) where x is THE VALUE OF x that maximizes, and P(x) is that maximum value of the product.

Then just find y by y=4-x.

:)

Answer 19

The answer is 4. Am i right? I solved it, i'm sorry for the late reply. Our net is so slow. :) thank you for the patience

Answer 20

@DebbieG

Answer 21

are you still there @DebbieG ?

Answer 22

i mean 2 is the answer

Answer 23

Sorry - yes, the answer is x=2, which gives you y=4-2=2.

Sum of 2, product is maximized at xy=4. :)

Answer 24

oops.... I mean sum of 4 :)

Answer 25

Yehey! Thank yooou so much. :) uhmmm, there is still one more question @DebbieG i need your help. The question i asked before are still the same but it is now a minimum. Here's the problem: Find two real numbers whose sum is 24 and whose product is a minimum. That's the problem :) this will be the last question, i really need your help. Please?

Answer 26

my answer is 12. Am i right?

Answer 27

Are you sure it doesn't say POSITIVE real numbers?

Answer 28

I'm not really sure about my answer. :3

Answer 29

Following the same process as before, you get x+y=24 so y=24-x

Substitute in the product P(x)=x(24-x)=\(24x-x^2\)

The problem here is \(P(x)=24x-x^2\) is downward opening, it has a max but NOT a min.

If it said "nonnegative real numbers" then you can take x=24, y=0. Then xy=0

But without that restriction, I could take, for example, x=-100, y=124. Then the product xy=-12,400 <0.

And so on, you can make the negative number as small as you want! Then the positive number is just that + 24. So there is no solution... because the quadratic does not have a minimum.

Answer 30

hmm. I see, thank yooou so much @DebbieG :) you really helped me a lot. Thank yooou so much ^_^v

Answer 31

You're welcome, happy to help! :)