Fermat's principle indicates the characteristic that the actual optical path always takes an extremum. Based on this, it is possible to further determine the specific type of extremum. Single-spherical-surface imaging, as a typical example of curved-surface imaging, has a constant actual optical path in paraxial ideal imaging. However, for the non-paraxial case, the type of extremum of the actual optical path has not been specifically discussed in the literature. This paper uses the mathematical method of finding higher-order derivatives to focus on discussing the type of extremum of the actual optical path corresponding to a given image point after reflection or refraction of light rays emitted from a given object point on the principal axis under non-paraxial conditions, deepening the understanding of non-paraxial imaging and Fermat's principle.