Proof of the Conditions for a Turning Value of f (x, y) without the use of Taylor's Theorem

Abstract

The standard method of establishing rigidly the tests for a maximum or minimum value of a function of two independent variables, depending as it does on the use of Taylor's Theorem and on a very critical consideration of the Remainder in that theorem, presents difficulties so considerable that it is not surprising that most text-books on the Calculus frankly decline to enter on the discussion, and assume the necessity and sufficiency of the well known Lagrange's Condition. It is the object of this paper to show that a satisfactory proof of the tests may be given, from the purely geometrical standpoint, without recourse to Taylor's Theorem. The method requires only an elementary knowledge of the process of changing the independent variables in partial derivatives, and may therefore be introduced comparatively early in the Calculus course.