Mean Value Theorems
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The mean value theorem
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If [Equation goes here - download the original to see it.] is a function of x (continuous and) having a differential coefficient at each point between and , then there exists a value of x between a and b such that[Equation] Proof This can be arranged as [Equation] and illustrated by the following graph [Diagram goes here - download the original to see it.] The graph is an intuitive proof of the theorem, which states that there is a point between A and B where the gradient of the tangent is equal to the gradient of the cord AB. A more rigorous proof proceeds via Rolle’s theorem. Define a function [Equation] as follow [Equation] Since [Equation] is continuous between a and b so is [Equation]. We have [Equation] and [Equation], so that [Equation] satisfies Rolle’s theorem. Hence there exists a value [Equation] between a and b such that [Equation] Differentiating (1) gives [Equation] On substituting [Equation] and the result [Equation] [Equation] Whence [Equation]. Corollary 1 If [Equation] for all x in [Equation], then [Equation] is constant for [Equation]. Proof Let [Equation] be points such that [Equation]. Let [Equation]. Then by Rolle’s theorem [Equation] Corollary 2 If [Equation] for all x in [Equation], then [Equation] is strictly increasing on the interval [Equation]. Proof Let [Equation] be points such that [Equation]. Let [Equation]. Then by Rolle’s theorem [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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