ABSTRACT
Under certain conditions a continuous function f: ℝn → ℝm can be approximated at each point x in ℝn by a linear function df(x):ℝn → ℝm, known as the differential of f at x. In the same way the differential df may be approximated by a bilinear map d2f(x). When all differentials are continuous then f is called smooth. For a smooth function f, Taylor’s Theorem gives a relationship between the differentials at a point x and the value of f in a neighbourhood of a point. This in turn allows us to characterise maximum points of the function by features of the first and second differential. For a real-valued function whose preference correspondence is convex we can virtually identify critical points (where df(x) = 0) with maxima.
