# Thread: Maximum and minimum of function

1. ## Maximum and minimum of function

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 .
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

2. ## Re: Maximum and minimum of function

Originally Posted by token22
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 .
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

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.

3. ## 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)

4. ## 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 ?