Given three positive numbers $x, y, z$ so that $xyz= 1, xy+ yz+ zx= 5$. find the maximum value $$x^{c}+ y^{c}+ z^{c}$$ for $c\geqq -1$ .
I think that the equality occurs at $x= y= 2, z= 1\div 4$ and its cyclic permutation. from hypothesis : $$5= xy+ yz+ zx\geqq 2\sqrt{x}+ \frac{1}{x}\therefore 2.4\geqq \sqrt{x}\geqq 0.5$$ from that, we have $$x^{c}+ y^{c}+ z^{c}\geqq x^{c}+ \frac{2}{x^{c}}$$ after that, we can use weight a.m.-g.m. to determine $c= constant$ but this way's only true for $c< 0$
I need to the help