Prove that there exists no maximum for the following?

Question

Let xyz = 1 for positive x, y, z. Show that min [x+y, x+z, y+z] has no maximum value.

The question comes from Mandelbrot #1, I have tried applying certain classical inequalities and have not gotten anywhere.

Answer 1

Let $x=y\rightarrow+\infty$. Thus, $$\min\{x+y,x+z,y+z\}=y+z\rightarrow+\infty.$$

Answer 2

You could make two of them as large as you want and make the constraint satisfied by using the remaining variable.