Convex Functions and Taylor’s Theorem

Abstract

In this chapter we consider the higher derivatives of a function f. These are f′′=(f′)′,$$f^{{\prime\prime}} = (f^{{\prime}})^{{\prime}},$$f(3)=(f′′)′,$$f^{(3)} = (f^{{\prime\prime}})^{{\prime}},$$etc. We extend the Mean Value Theorem to an analogous statement about the second derivative, and this takes us naturally to the notion of convexity. Once there, we meet the very important Jensen’s Inequality. Then we extend the Mean Value Theorem to the (n+1)st derivative—this is Taylor’s Theorem. We prove that e$$\mathrm{e}$$is irrational and we take a brief look at Taylor series. Taylor’s TheoremTaylor series