Symmetry and a bit of experimentation lead to the guess that the minimum is achieved for x=y=z, and so the minimum value is conjectured to be 9/2.
To confirm this conjecture it is sufficient to show that 9/2 is a lower bound. One neat way to do this involves the Arithmetic-Geometric-Mean inequality, which for three positive real variables a, b, and c states that:
This inequality can be applied twice with appropriate choices for a, b, and c in terms of x, y, and z to show that the expression given is indeed greater than or equal to 9/2.
(The notation >= for `greater than or equal to' is used to overcome limitations of some net browsers.)