Mean Value Theorems
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Mean value theorem for second derivatives
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If the functions [Equation goes here - download the original to see it.] and [Equation] have second differential coefficients such that [Equation] and [Equation] do not vanish simultaneously in the interval [Equation], and if [Equation] is not zero, then there is a number [Equation] between [Equation] for which [Equation]. Proof. Let [Equation] be the function defined by [Equation] where A, B and C are constants such that [Equation] [Equation] [Equation]. By Rolle s theorem, since [Equation], there is a number [Equation] between [Equation] such that [Equation]. Also by Rolle s theorem, since [Equation] there is a number [Equation] between [Equation] such that [Equation]. We now apply Rolles theorem to the function [Equation], which vanishes when [Equation]. Thus, there is a number [Equation] between [Equation], and hence also between [Equation], such that [Equation]. Hence [Equation] But [Equation] by hypothesis, so we can divide by [Equation] to obtain [Equation] We may also obtain from the three equations for A, B and C [Equation] And since [Equation] by hypothesis [Equation] Hence [Equation] Application of mean value theorem to limits If [Equation] and [Equation] are (continuous) functions such that [Equation] and if [Equation] both exist, with [Equation] then [Equation]. Proof Since [Equation] we have [Equation] Then, as [Equation] we have [Equation] Hence [Equation] Corollary If [Equation] and [Equation] are (continuous) functions such that [Equation], and if [Equation] then [Equation] Proof Proof is by application of the Cauchy mean value theorem, to show that [Equation] for some number [Equation] between x and a. As [Equation] then [Equation] also. Hence [Equation]
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Contents of
Mean Value Theorems
1 Rolles theorem
2 The mean value theorem
3 Cauchy’s mean value theorem
4 Mean value theorem for second derivatives
5 Maxima and minima
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