Find min

universe · 2011-04-02 21:05:48

Hi everybody, can you solve this problem?
Find the minimum value of the expression:

where are positive real numbers satisfying:

Last edited by universe (2011-04-02 21:08:01)

bobbym · 2011-04-03 02:56:37

Hi universe;

Please ignore this post and see the later posts for the correct answer.

Welcome to the forum. The method of Lagrangian Multipliers failed to produce a minimum. Also I was unable to utilize the AMGM. But I could find and experimentally good answer which I believe is the minimum or very, very close to it.

The minimum occurs at

It produces a minimum of:

gAr · 2011-04-03 03:18:12

Hi bobbym and universe,

bobbym, I too tried the Lagrangian Multipliers and left that ugly expression.
Do positive numbers incIude zero?
I substituted your values in the equation, it yields about 933.936.
I think minimum occurs at (1,1,1), which yields 648.

Last edited by gAr (2011-04-03 03:19:00)

bobbym · 2011-04-03 03:22:13

Hi gAr;

I am getting for my numbers a minimum of about 9.25. And no, ( to your question, 0 is not a positive number ) that is the problem I am hoping that he would see. I can make the minimum as small as I want by approaching 0. Therefore it does not exist.

gAr · 2011-04-03 03:40:00

Hi bobbym,

I double-checked:

I'm confused how you got your value!
And mathematica showed (1,1,1).

bobbym · 2011-04-03 03:41:48

Hi gAr;

I am double checking again, please hold on. Could be a mistake, thanks for checking.

Okay, I am getting for:

With my values. Which does not match yours. I do not know what is happening here!!!!!!

gAr · 2011-04-03 03:50:46

You checked only for one term, what about other two?!

bobbym · 2011-04-03 03:54:14

Hi gAr;

I see it now!!!!!

gAr · 2011-04-03 03:59:14

Now I understand,

I'll be back in a few moments, see you...

bobbym · 2011-04-03 04:00:55

Thanks gAr!!!!

I went blind there and did not notice that. I can only say that while I was in combat I was in hit by a mortar shell squarely in the head. It blew most of my cranium away and my eyes too. The medics used a medical vacuum to drag up the remnants of my brain and eyes. They were reattached in a record setting 15 hour surgery. They sometimes do not work correctly.

I will reapply the method of Lagrangian multipliers.

Okay, got it to work. Using the method of Lagrangian multipliers and a computer to solve the simultaneous set on non linear equations the minimum is:

a = 1, b = 1, c = 1 for a value of 648. ( did not gAr already say that? ). ( Correction by gAr too! )

Now if you are posing this as a teaser ( or a homework assignment or contest ) then it belongs in puzzles and games and the above answer will be totally unacceptable. But since you are posing this in help me then this answer will be helpful.

gAr · 2011-04-03 04:57:19

Hi bobbym,

No problem.

a = 1, b = 1, c = 1 for a value of 624. ( did not gAr already say that? ).

I said 648

You could solve it? Cool!
Did you try it in paper too?

This question may be for extra credit.

bobbym · 2011-04-03 05:05:06

Hi gAr;

I am afraid to even look at this problem now! You are right I got 648 like you did and wrote 624.

It is not too difficult to set up the problem using Lagrange but it is all but impossible to solve the equations by hand. I used Mathematica for that part. I now think that this is best done by inequalities.

This question may be for extra credit.

If I thought that I would be reluctant to even work on the problem. Extra credit is earned by doing it yourself.

gAr · 2011-04-03 05:14:22

Ok, I spent much time trying this. Could guess the answer by symmetry, but proving it proved tough!

I am just telling it cannot be a normal homework problem, creepy problem!

bobbym · 2011-04-03 05:24:02

I agree, it looks like an olympiad problem actually. But it could be posed as a contest problem too. Some sites do that. I usually check around before answering any question. I do not want to assist someone in cheating.

These are our guidelines from MIF for answering questions, we are supposed to stick to it. I know you cannot be rigid in following rules but you cannot bend them too much either.

http://www.mathisfunforum.com/viewtopic.php?id=14654

But in this case I felt it was a legitimate question.

Jane is our expert on inequalities and I have not seen her in a while.

gAr · 2011-04-03 05:37:24

Oh, ok.

Jane is indeed an expert, I have seen her solutions, amazing!

I have seen you solve this one: http://www.mathisfunforum.com/viewtopic.php?id=9076
What did you use for that problem?

bobbym · 2011-04-03 05:52:35

Hi gAr;

I wish I could remember. I am looking at my notes for inequalities and this problem is not there. Tony123's problem are putnam or olympiad level so they are all very tough. I will work on it again and see if I can stumble onto some solution.

gAr · 2011-04-03 05:59:12

Ok.

Wolfram solves it in a snap, how brilliant!
I'll take some rest, see you later.

bobbym · 2011-04-03 16:22:14

Hi gAr;

I was able to recreate the method I have added to the original post here:

http://www.mathisfunforum.com/viewtopic.php?id=9076

I agree, it is amazing how Wolfram can get this answer in the blink of an eye.

gAr · 2011-04-03 16:51:43

Hi bobbym,

Thanks for posting the solution.

I was reading about inequalities. Looks like it should be used for this problem, like you already said.

bobbym · 2011-04-03 17:11:28

Except for one problem. The expression to be maximized is not an inequality. That makes it hard to use the AMGM on it. What are we supposed to put on the RHS?

gAr · 2011-04-03 17:30:12

Hmmm,

I haven't done problems on inequalities, but I feel the product might be easier to minimize.

bobbym · 2011-04-03 18:00:40

That is a possibility. We do not have to work on minimizing it anymore. We can work on it as an inequality. I will try that.

gAr · 2011-04-03 18:05:12

Ok, I'll also try that way.

bobbym · 2011-04-03 19:46:07

Hi gAr;

As soon as you set up the inequality.

You know that equality occurs only when a = b = c all other values are higher so that is the minimum. Change the constraint to be

a = 1 so b = 1 and c = 1 and we are done.

gAr · 2011-04-03 20:01:47

Hi bobbym,

Yes! Thank you.

Math Is Fun Forum

#1 2011-04-02 21:05:48

Find min

#2 2011-04-03 02:56:37

Re: Find min

#3 2011-04-03 03:18:12

Re: Find min

#4 2011-04-03 03:22:13

Re: Find min

#5 2011-04-03 03:40:00

Re: Find min

#6 2011-04-03 03:41:48

Re: Find min

#7 2011-04-03 03:50:46

Re: Find min

#8 2011-04-03 03:54:14

Re: Find min

#9 2011-04-03 03:59:14

Re: Find min

#10 2011-04-03 04:00:55

Re: Find min

#11 2011-04-03 04:57:19

Re: Find min

#12 2011-04-03 05:05:06

Re: Find min

#13 2011-04-03 05:14:22

Re: Find min

#14 2011-04-03 05:24:02

Re: Find min

#15 2011-04-03 05:37:24

Re: Find min

#16 2011-04-03 05:52:35

Re: Find min

#17 2011-04-03 05:59:12

Re: Find min

#18 2011-04-03 16:22:14

Re: Find min

#19 2011-04-03 16:51:43

Re: Find min

#20 2011-04-03 17:11:28

Re: Find min

#21 2011-04-03 17:30:12

Re: Find min

#22 2011-04-03 18:00:40

Re: Find min

#23 2011-04-03 18:05:12

Re: Find min

#24 2011-04-03 19:46:07

Re: Find min

#25 2011-04-03 20:01:47

Re: Find min

Board footer