Real and Generic Data without Unconstrained Best-Fitting Verhulst Curves and Sufficient Conditions for Median Mitscherlich and Verhulst Curves to Exist

A commonly taught scientific method for building mathematical models uses finite computations to approximate the curve of a specified type that best fits the data, without checking whether any such best-fitting curve exists: not every regression objective need have a global unconstrained minimum. One counterexample will confute its theoretical foundation: any triple of points with super-exponential growth does not admit of any unconstrained bestfitting Verhulst logistic curve, regardless of the regression criterion. Moreover, because the set of all such triples is open, there are still no best-fitting Verhulst curves after sufficiently small but arbitrary perturbations of the data. Nevertheless, the present explanations show that through each triple of points with sub-exponential growth passes a unique Verhulst curve. Furthermore, if every triple of data grows sub-exponentially, then for the reciprocal data there exists a median Mitscherlich curve whose reciprocal is a Verhulst curve. Applications range from alchemy to zoology.