Question

the inability to produce a gag reflex could indicate a problem with the facial nerve (vii) glossopharyngeal nerve (ix) abducens nerve (vi) hypoglossal nerve (xii)

Explanation:

The gag reflex is triggered by stimulation of the posterior pharynx. The glossopharyngeal nerve (IX) is responsible for the sensory input (afferent limb) of the gag reflex. The facial nerve (VII) is mainly for facial muscles and taste in the anterior tongue. The abducens nerve (VI) controls eye movement. The hypoglossal nerve (XII) controls tongue muscles. So, a problem with the glossopharyngeal nerve (IX) would affect the gag reflex.

Answer:

glossopharyngeal nerve (IX)