To determine whether the unknown bacterium is a chemotroph, the most appropriate method would be:
growing the bacterium in a sulfur solution to see if it breaks the inorganic molecules down.
Chemotrophs obtain energy from the chemical bonds of inorganic or organic molecules, rather than from sunlight (as phototrophs do). By observing if the bacterium can break down inorganic molecules like sulfur, you would be able to determine if it is obtaining energy through chemotrophic means.