Why does the profit function in standard neoclassical theory have exactly one maximum?
Mathematically, most neoclassical  models assume that the profits are concave. This guaranties the uniqueness of the maximum. In economic terms, the Neoclassical theory usually assumes that the law of diminishing returns holds. Thus, the more you hire workers, the more you produce, but at a diminishing rate. 

I am not so sure what your professor has in mind when he brings up the "static nature of the theory". There are many dynamic neoclassical models and most of them have a unique maximal profit that also arises from having an appropriate form of diminishing returns, i.e. concavity of the profit function.