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#1 2014-10-08 14:58:26
Maximise ...
If x + y + z = 3, and x, y, z are non-negative real numbers:
What is the maximum value of
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#2 2014-10-08 17:34:49
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
How is that bracketed on those exponents?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#3 2014-10-08 19:27:44
Re: Maximise ...
2^(x y) + 2^(y z) + 2^(x z)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#4 2014-10-08 19:48:27
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
Re: Maximise ...
hi Agnishom,
Why?
It would make no sense for the maximum to be biased towards either x, or y, or z. So It ought to be when they are equal.
This may not be correct though. 
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#5 2014-10-08 21:06:02
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
Hi;
I have latexed it up for you.
The method of Lagrangian multipliers will give you the answer you want.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#6 2014-10-08 22:07:26
- Olinguito
- Member
- Registered: 2014-08-12
- Posts: 649
Re: Maximise ...
Why?
It would make no sense for the maximum to be biased towards either x, or y, or z. So It ought to be when they are equal.
That was what I thought as well until I found this: 
That might be the maximum but I don’t know how prove it yet (with or without Langrangian multipliers). 
Bassaricyon neblina
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#7 2014-10-08 22:38:09
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
Hi;
You have found the maximum value.
This makes me remember an amusing story, did anyone know that Lagrange was called Bubba by his closest friends? Yes he was. I remember sitting in a pub in New Orleans with Bubba and he said, "bobbym I have discovered a truly amazing method to solve maxima-minima problems." I asked what it was and he explained it wonderfully, then he said he was going to call it Bubbanian Multipliers. It took me all night and about 15 tall ales to convince him to call it the name it now bears, "Lagrangian Multipliers."
Anyway, the methods of EM are more powerful than even he could know and made short work of this problem. I am getting:
as the maximum value.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#8 2014-10-09 00:44:22
Re: Maximise ...
Can you show that it is the maximum value in atleast two ways?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#9 2014-10-09 02:33:42
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
I only did it in one way. Sometimes, we do not have the luxury of having two equally good methods. The answer given is one I am willing to bet on, meaning the odds are in my favor. I would also like the history of the question.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#10 2014-10-09 03:28:57
Re: Maximise ...
Please do so
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#11 2014-10-09 03:31:35
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
The history please?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#12 2014-10-09 04:49:03
Re: Maximise ...
Sorry, I misread the post.
I got the question from a friend. From where he got it, I do not know.
Last edited by Agnishom (2014-10-09 04:51:37)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#13 2014-10-09 05:11:43
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
Okay, so what is it you need?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#14 2014-10-09 06:15:10
Re: Maximise ...
The maximum value of the expression and some argument that will make me believ it
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#15 2014-10-09 06:32:16
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
The maximum value is
The first and best way because it got there is:
1) Determine the maxima numerically to as many digits as possible ( about 30 ).
2) Turn the numerical constant into a nice closed form by using integer relation or PSLQ algorithms.
I have gone over these concepts with you perhaps a dozen times or more so I am sure you can follow.
The second way is to solve the problem using the method of Lagrangian Multipliers. This produced a bunch of tough equations that might be solved but I abandoned the whole idea in favor of the first.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#16 2014-10-09 12:06:10
Re: Maximise ...
How do you do 1)?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#17 2014-10-09 16:41:53
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
Hi;
You use a computer to do the numerical work of course. Then a PSLQ is brought to bear.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#18 2014-10-09 19:23:35
Re: Maximise ...
What is the algorithm?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#19 2014-10-09 19:26:04
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
Algorithm, schmalgorithm...
Want the code?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#20 2014-10-09 19:31:47
Re: Maximise ...
Okay
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#21 2014-10-09 19:35:39
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
NMaximize[{2^(x*y) + 2^(x*z) + 2^(y*z), x + y + z == 3, x >= 0,y >= 0, z >= 0}, {x, y, z}, WorkingPrecision -> 60]Can you do the rest?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#22 2014-10-09 19:51:54
Re: Maximise ...
Yes.
Can you tell me how to code up that fucntion myself?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#23 2014-10-09 19:59:01
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
You can now get the analytical answer?
Code it up? I do not understand.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#24 2014-10-09 20:00:06
Re: Maximise ...
What good is the analytical answer?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#25 2014-10-09 20:04:41
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Maximise ...
What good is the analytical answer?
The analytical answer is interesting and useful in its own way. It is easier for us humans to spot patterns and laws when we see analytical answers. Also, when it is correct ( and when is a big word ) it is an exact answer. In a sense this technique ties together classical and numerical math.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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