October 8th, 2011, 10:19 AM  #1 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  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) 
October 8th, 2011, 02:18 PM  #2 
Newbie Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0  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 .

October 8th, 2011, 02:56 PM  #3 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  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) 
October 8th, 2011, 04:05 PM  #4 
Newbie Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0  Re: Maximum and minimum
What I meant was that you wrote this which is (nonsense 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. 
October 8th, 2011, 05:17 PM  #5 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  Re: Maximum and minimum
Ups, sorry i didn't notice it. it is supposed to be .

October 9th, 2011, 06:51 AM  #6 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  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 AMGM 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) 
October 9th, 2011, 09:12 AM  #7 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  Re: Maximum and minimum
*SOLVED*

October 9th, 2011, 01:50 PM  #8 
Newbie Joined: May 2011 Posts: 28 Thanks: 0  Re: Maximum and minimum
Sorry, using AMGM inequality i know that it has maximum.


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