# Maximum and minimum of function

• Oct 8th 2011, 10:37 AM
token22
Maximum and minimum of function
Hello, could you help how to solve this:
Determine if there exists minimum or maximum of function http://latex.codecogs.com/gif.latex?.....x_%7Bn%7D%7D with condition http://tiny.cc/41ta1.
I know how to find out extrems (using Langrangeov function,..), but n variables somehow confuse me.

Thanks

edit
I hope I put it to right category
edit2
okay seems I'll never get answer in this forum
• Oct 9th 2011, 10:34 AM
Opalg
Re: Maximum and minimum of function
Quote:

Originally Posted by token22
Hello, could you help how to solve this:
Determine if there exists minimum or maximum of function http://latex.codecogs.com/gif.latex?.....x_%7Bn%7D%7D with condition http://tiny.cc/41ta1.
I know how to find out extrems (using Langrangeov function,..), but n variables somehow confuse me.

Thanks

edit
I hope I put it to right category
edit2
okay seems I'll never get answer in this forum

If the original post had not had those confusing commas you might have had a reply sooner.

You don't need Lagrange multipliers for this, it just needs the AM–GM inequality, which tells you straight away that the max occurs when \$\displaystyle x_1 = x_2 = \ldots = x_n = c/n.\$ There is no minimum, because you can get arbitrarily close to (but not equal to) zero by taking one of the x's small enough.
• Oct 9th 2011, 01:38 PM
token22
Re: Maximum and minimum of function
Hm, strange beacuse i used Lagrange where I used partial derivates the whole function (with n variables and so on) - Lagrange function that shoud be equal to 0 but it gave me that x_1,..x_n should be equal to zero. So I thought it has no extrems.

Edit
Yes this solution with AM-GM inequality looks good.
Thanks (I found it earlier but couldn't use it right)
• Oct 9th 2011, 10:54 PM
token22
Re: Maximum and minimum of function
By the way, will this solution work for just sum of x equal to c? Souldn't be condition equal to c/n ?