Optimization, finding two numbers whose sum is minimal

[Femme_physics]

Homework Statement

From all positive numbers x and y that hold y(x+2) = 9 , find the two numbers whose sum x+y is minimal

The Attempt at a Solution

Attached. My idea here is to take the derivative of y with respect to x, and set it equal to zero. This is how I understand you solve for optimization problems and find minima and maxima. Have I set the derivative correctly? I think my idea is correct, but the way I practically approach it isn't.

Will appreciate any hints and directions I can get :)

 

[jhae2.718]

You want to minimize f(x,y) = x + y; how can you write f(x, y) = x + y as a function of one variable?

 

[Metaleer]

Be careful, you are not trying to optimize the constraint itself, but rather another function subject to the constraint.

You have two ways to solve it.

You are trying to find the minimum vale of $$S(x, y) = x+y$$, subject to the constraint $$y(x+2) = 9$$. So one way would be to solve for $$y$$ in the constraint and plug this value into $$S(x, y)$$, from which you get a function of $$x$$, which you optimize using differential calculus as you have tried in your attempted solution.

The second way would be to use Lagrange multipliers - I assume you have not taken multivariable calculus yet, so I will not elaborate to confuse you. However, if you have taken multivariable calculus (or are covering it now) or wish to see how the Lagrange multiplier method works, just say it and I'll explain how it's used.

Best of luck. :)

 

[I like Serena]

So the function you would try to optimize is:

$$f(x)=x + \frac 9 {x+2}$$

Can you take the derivative of that?

And solve it for being equal to zero? That is: f'(x) = 0

 

[AlexChandler]

So the function you would try to optimize is:

$$f(x)=x + \frac 9 {x+2}$$

Can you take the derivative of that?

And solve it for being equal to zero? That is: f'(x) = 0

Looks like the right idea to me.

 

[Femme_physics]

jhae - hint level 1
Metaleer - hint level 2
I Like Serena - hint level 3
Alex - confirms hint level 3

I love you guys ;)


Reading all the replies gave me the clue in. So basically, I try to minimize x+y, and I was given a function that is my constraint. So, all I have to do is rewrite the y in the formula as 9/x+2, that being the constraint, like ILS suggested. Aha! So I did have the right idea of how to solve it!

I'm off the scanner for today, but setting the derivative to zero I did get x=1 and x=-5. Since they said only real numbers, I remove x=-5. Plugging it into the original formula I get y=3. So, answers are, x=1 and y=3

That's true according to the manual! :)

Wait there's one more thing, they're asking for the minimal value of x+y... er...what does it mean? Should I just do... 1+3 = 4?

Because the answers manual says the answer to this question is indeed 4...but it seems like a stupid question.


Anyway, thanks for helping me see it through! :D

The second way would be to use Lagrange multipliers - I assume you have not taken multivariable calculus yet, so I will not elaborate to confuse you.

*head explodes at "Lagrange"*


PS I tried using LaTeX but fudged it. Where is the fraction function?

 

[AlexChandler]

PS I tried using LaTeX but fudged it. Where is the fraction function?

To make a fraction, you can just type

\frac{numerator}{denominator}

But don't type "numerator" and "denominator", just put whatever expression you want to go there.

Also, you can click on any LaTeX that you see somebody else has done, and it will show you the code that created it.

 

[I like Serena]

I'm off the scanner for today, but setting the derivative to zero I did get x=1 and x=-5.

Since they said only real numbers, I remove x=-5. Plugging it into the original formula I get y=3. So, answers are, x=1 and y=3

That's true according to the manual! :)

They said "only positive numbers", so your conclusion is correct.
(However note that "real numbers" can also be negative.)

Wait there's one more thing, they're asking for the minimal value of x+y... er...what does it mean? Should I just do... 1+3 = 4?

Because the answers manual says the answer to this question is indeed 4...but it seems like a stupid question.

*pout* it's not stupid

But yes, that is correct

PS I tried using LaTeX but fudged it. Where is the fraction function?

The fraction function is "\frac".

So you'd type for instance: [ tex ] y = \frac {9} {x+2} [ /tex ]
But without the spaces around "tex" and "/tex".
That way you'll get this:

$$y = \frac {9} {x+2}$$

*oooohhh* it's *magic*

Click on it and you'll get a popup window with more information and with a link to quick reference documentation.

 

[Femme_physics]

They said "only positive numbers", so your conclusion is correct.
(However note that "real numbers" can also be negative.)

Oh, right. You're always picking up on my slips, I like that :)

But yes, that is correct

Neat ^^

To make a fraction, you can just type

\frac{numerator}{denominator}

But don't type "numerator" and "denominator", just put whatever expression you want to go there.

Also, you can click on any LaTeX that you see somebody else has done, and it will show you the code that created it.

Click on it and you'll get a popup window with more information and with a link to quick reference documentation.

Sorry-- I mean when I click the "Show/Hide Latex Reference" buttom. Where do I click next? "Boundaries, Logical, Sets"... I don't find the fraction function, it's like finding a needle in a hay sack.

 

[I like Serena]

Sorry-- I mean when I click the "Show/Hide Latex Reference" buttom. Where do I click next? "Boundaries, Logical, Sets"... I don't find the fraction function, it's like finding a needle in a hay sack.

I've lost you.
Where did you find this "Show/Hide Latex Reference" buttom, or this "Boundaries, Logical, Sets"?

The quick reference I meant is here:
https://www.physicsforums.com/misc/howtolatex.pdf

On the first page it says:

Fractions. Use \frac to display fractions. Example: \frac{\pi^2}{6} gives $$\frac{\pi^2}{6}$$ .

[edit]Hold on, I understand. That's in the advanced edit mode. That one is useful too to find stuff, but it'll take awhile to take it in. I think I'd start with the quick reference.[/edit]

[edit2]And to answer your question, the \frac is in the "Math" section.[/edit2]

 

[Femme_physics]

Ah, yes! That's it. Not used to those symbols as I've never written with it before. I'll try to get the hang of it for the sake of our lovely helpers and you. :) Though, I'll also scan when I can. Thanks!

 

[Mark44]

Ah, yes! That's it. Not used to those symbols as I've never written with it before. I'll try to get the hang of it for the sake of our lovely helpers and you. :) Though, I'll also scan when I can. Thanks!

Personally, I like to see LaTeX better than I like to see scanned material, because 1) the scanned material doesn't appear when you type a response, and 2) one can insert a comment at the precise point where the work went awry.

 

[Femme_physics]

Hmm... Good point, Mark44. As long as my work isn't too long I'll consider LaTeX from now on then. Cheers :)