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October 8th, 2011, 11:19 AM   #1
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Maximum and minimum

Hello, could you help how to solve this:
Determine if there exists minimum or maximum of function \mathbb{R}^{+})^{n} \rightarrow \mathbb{R}, f(x_{1},...,x_{n}) = \sqrt[n]{x_{1},...,x_{n}}" /> with condition . I know how to find out extrems (using Langrangeov function,..), but n variables somehow confuse me.

Thanks (though i do not think somebody reply)
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October 8th, 2011, 03:18 PM   #2
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Re: Maximum and minimum

The way you define must be flawed, as you first state that it takes values in . Were you trying to write ? In that case does not attain any maximum value, nor minimum if you only define it for . Yet and .
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October 8th, 2011, 03:56 PM   #3
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Re: Maximum and minimum

Well, I just rewrite one of my Problems that i got to solve. Function is thus given as I wrote R^{+})^{n} \rightarrow \mathbb{R}" /> . I think it just take only positive real numbers. Function is with n variables and condition is as I wrote (but i think you know i I meant it). I forgot to think about series, so if I could write it as you did (to serie) it just could be fine to show it diverges, right? If I didn't write answer you expected, please tell me, I would like to solve it.
(by the way, my lecturer has tend to create quite strange Problems)
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October 8th, 2011, 05:05 PM   #4
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Re: Maximum and minimum

What I meant was that you wrote this



which is (non-sense to me) inconsistent with . I thought perhaps you meant



which is unbounded from infinity, but bounded away from zero by the constraint that only maps the strictly positive part of .

Therefore, it does not have a minimum or maximum.
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October 8th, 2011, 06:17 PM   #5
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Re: Maximum and minimum

Ups, sorry i didn't notice it. it is supposed to be .
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October 9th, 2011, 07:51 AM   #6
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Re: Maximum and minimum

Okay, i solved it as I took a few variables than just made Lagrange function - it gave me that must be not zero otherwise it is not equal to zero. I am not sure if it is fine.
Also, somebody told me to look up AM-GM inequality, but i am not sure how this can help me. What do you think?
(btw I am starting to think that I made thread in wrong category)
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October 9th, 2011, 10:12 AM   #7
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Re: Maximum and minimum

*SOLVED*
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October 9th, 2011, 02:50 PM   #8
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Re: Maximum and minimum

Sorry, using AM-GM inequality i know that it has maximum.
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