The best explanation of the role of a city under central place theory is:
c a city exists above a threshold and under a range of economic production as it provides for its surrounding market area.
Central place theory, developed by Walter Christaller, suggests that cities (central places) exist to provide goods and services to the surrounding rural areas in a hierarchical manner, with larger cities serving larger areas and providing more specialized services. Thus, option c captures the concept of cities existing to serve their hinterlands while adhering to the principles of the theory.