Let $x,y, z$ be reals such that $x^2+y^2+z^2=3$. What is the minimum value of $2^{1/x}+2^{1/y}+2^{1/z}$?
First I thought taking $x,y,z$ as $-1$ which leads to $3/2$. However by trial and error, if I take $x=y=-1.1$ then $z=-\sqrt{0.58}$. In that case I get a value of the expression $1.468<1.5$, so clearly what I took first is wrong.
So how to go about solving this?