Is it true that Maxima/Minima of symmetric polynomial functions in $x,\frac{1}{x}$ lies when $x=\frac{1}{x}$,
For example,
$$x^2+\frac{1}{x^2}$$ Has it's Minima at $x=1$
We may also write$$f(x)=\sum_{r} k(x^r+x^{-r})$$
Is it also true when $x$ is complex, for eg. Maxima of $$|x+\frac{1}{x}|$$
It is possible to construct examples of such functions that do not have extrema at $x = \pm 1$. For instance, by exploiting the continuous function $$ g(x) = \cases{0 & if $x = 0$\\x\sin(1/x) & otherwise} $$ This function doesn't behave nicely around $x = 0$. We can move that non-nice behaviour to $x = \pm 1$ by considering, for instance, $g(|x| - 1)$. Then make it symmetric with respect to $x\mapsto 1/x$ by simply adding another copy with $1/x$ in place of $x$. The final result is $$ f(x) = g(|x| - 1) + g\left(\left|\frac1x\right| - 1\right) $$ and this does not have extreme values at $x = \pm 1$.