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In this problem

minimize a function using AM-GM inequality

we discussed about the minimum points of a simmetric function.
Now I would like to ask to you this:

If you have a rational function $f(x,y)$ that is simmetric in the two variables, meaning $f(x,y)\equiv f(y,x)$, then is it always true that the point of minimum/maximum, if there exist, are such that $x=y$?

In the case $f(x,y)=g(x)+g(y)$, the partial derivatives tell me that there exist critical point with $x=y$, but are they of minimum/maximum? and are thay all the minimums/maximums?

And in the case $f(x,y)=g(x)g(y)$ or, in general, when you can "saparate" in some way the two variables?

And when there's a simmetric polynomial constraint $h(x,y)=c\in\mathbb{R} $?

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Exodd
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    Symmetric problems have symmetric solution sets, but it's not always the case that the individual solutions are symmetric. E.g., maximize $x^2+y^2$ subject to $|x|+|y|\le 1$. (Or, subject to $(x+y)^{100}+(x-y)^{100}\le 1$ if you insist on polynomial constraints.) –  Oct 30 '14 at 00:12
  • @StevenTaschuk can you find some example with a finite number of minimums non simmetric? – Exodd Oct 30 '14 at 01:42
  • Minimize $x^2+y^2$ subject to the constraint that $(x,y)$ lie on the boundary of a regular octagon with vertices $(\cos\theta,\sin\theta)$ for $\theta=\frac{2\pi n}{8}$. –  Oct 30 '14 at 01:44
  • Or, take the square $[-1,1]^2$ and cut a tiny bit off at each of the corners, and optimize over the boundary. Then the minimum of $x^2+y^2$ is at $(\pm 1,0)$ and $(0,\pm 1)$ and the maximum is close to, but not exactly at, $(\pm 1,\pm 1)$. –  Oct 30 '14 at 01:48
  • Ok, you can play with boundary and find suh examples, but this time I'll try to be more specific: there exist a polynomial simmetric function defined on the unitary ball of $\mathbb{R}^n$ such that it has a finite number of non-simmetric minimums in the interior of the ball? – Exodd Oct 30 '14 at 02:09
  • Can't I just pick any desired minima $(m_i){i=1}^k$ and define $f(x) = \prod{i=1}^k |x-m_i|^2$ ? If the set of minima is symmetric under permutation of coordinates, this polynomial will be too. –  Oct 30 '14 at 02:13

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