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I'm wondering if the following argument is right:

"The optimum (minimum/maximum) value of a symmetric function f(x_1,x_2,...,x_n) (By 'symmetric' I mean that f remains same if we alter any x_i's with x_j's), if exists, should be at the point x_1=x_2=...=x_n".

Please help me by proving (or disproving, i.e, a counter example) this argument. I have examine some well-known symmetric functions and these obey the rules. However, a general proof (or reference to literature, document etc.) is needed.

[I think this may be right, in the sense that the graph of the function (in n+1 dimensional space) should be symmetric about all axes, so if f has optimum at x_1=a, by the symmetry of the curve, x_2=...=x_n=a.]

Regards,

NaturePaper