Final answer:
To determine if the oculomotor nerve is damaged, the patient should be asked to look up and down, tracking an object to test the function of the extraocular muscles controlled by the nerve.
Step-by-step explanation:
To determine if the oculomotor nerve (III) was damaged, the appropriate test would be to have the patient look superiorly and inferiorly. The oculomotor nerve controls most of the extraocular muscles as well as the muscle that lifts the eyelid. Damage to this nerve can result in an inability to move the eye in certain directions, ptosis (drooping of the eyelid), and problems with pupil constriction.
Testing eye movement involves having the patient track an object, like the tip of a pen, with their eyes. This tests the function of the extraocular muscles which are innervated by the oculomotor, trochlear, and abducens nerves, and their coordination by the medial longitudinal fasciculus (MLF). Observations of eye movement irregularities, like failing to move the eyes conjugately, could indicate an issue with one of these nerves, potentially revealing oculomotor nerve damage.