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 October 8th, 2011, 11:19 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 $f\mathbb{R}^{+})^{n} \rightarrow \mathbb{R}, f(x_{1},...,x_{n}) = \sqrt[n]{x_{1},...,x_{n}}" /> with condition $x_{1} + ... + x_{n}= c, c \in \mathbb{R}^{+} , x_{1},...,x_{n} > 0=$. 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, 03:18 PM #2 Newbie   Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0 Re: Maximum and minimum The way you define $f$ must be flawed, as you first state that it takes values in $\mathbb{R}$. Were you trying to write $f(\underline{x})= (\sum_{i=1}^n x_i)^{1/n}$ ? In that case $f$ does not attain any maximum value, nor minimum if you only define it for $\mathbb{R}_+^n$. Yet $\sup f= \infty$ and $\inf f= 0$.
 October 8th, 2011, 03:56 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 $fR^{+})^{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, 05:05 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 $^n\sqrt{x_1,\dots,x_n}$ which is (non-sense to me) inconsistent with $f: \mathbb{R}_+^n \to \mathbb{R}$. I thought perhaps you meant $^n\sqrt{x_1+\cdots+x_n}$ which is unbounded from infinity, but bounded away from zero by the constraint that $f$ only maps the strictly positive part of $\mathbb{R}^n$. Therefore, it does not have a minimum or maximum.
 October 8th, 2011, 06:17 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 $\sqrt[n]{x_{1}...x_{n}}$.
 October 9th, 2011, 07:51 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 $x_{1},..x_{n}$ 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)
 October 9th, 2011, 10:12 AM #7 Newbie   Joined: May 2011 Posts: 28 Thanks: 0 Re: Maximum and minimum *SOLVED*
 October 9th, 2011, 02:50 PM #8 Newbie   Joined: May 2011 Posts: 28 Thanks: 0 Re: Maximum and minimum Sorry, using AM-GM inequality i know that it has maximum.

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