Mean Value Theorems
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Rolles theorem
Equations are omitted for technical reasons - download the original pdf
Let [Equation goes here - download the original to see it.] be continuous on the closed interval [Equation goes here - download the original to see it.]. Let the curve [Equation] having a tangent on the open interval between a and b; that is, [Equation] on [Equation]. Then there exists at least one value of x between a and b at which [Equation] vanishes; there exists a c, with [Equation] such that [Equation].The following diagram illustrates the theorem and explains why it as geometrically transparent. [Diagram goes here - download the original to see it.] Proof. Let [Equation] and let [Equation]. Suppose [Equation] then [Equation]. Suppose [Equation], then by the Intermediate value theorem, there exists a c, where [Equation] such that [Equation]. We are given that [Equation] exists. Now suppose [Equation] then f is strictly increasing a c. Then there must be at least one d, where [Equation], such that [Equation]. Then [Equation]. Hence [Equation]. But likewise, [Equation] leads to a contradiction; 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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