Does the function attain a minimum on this domain?

1. Check nonnegativity, apply AM–GM and verify attainable equality.

2. Calculate or simplify this relation.

   $$x+\frac{64}x-16=\frac{(x-8)^2}{x}>0$$

3. Calculate or simplify this relation.

   $$x\to8^{+}\Rightarrow x+\frac{64}x\to16$$

4. An infimum need not be a minimum.

No; its infimum is 16, approached but not attained.

Checks and common pitfalls: An infimum need not be a minimum.