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

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.

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)

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 ?