The best explanation of the role of a city under central place theory is:
A city exists above a threshold and under a range of economic production as it provides for its surrounding market area.
This choice accurately reflects the idea that cities function as central places that serve a specific market area, offering goods and services necessary for the population within that area. Central place theory, developed by Walter Christaller, explains how and where central places (such as cities) are established, based on their ability to provide economic functions to the surrounding region.