Mean Value Theorems
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Maxima and minima
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Definition. A continuous function f is said to have a maximum at c if there is a neighbourhood of c such that [Equation goes here - download the original to see it.] provided [Equation]. f has an minimum if we substitute > for < in the above. Theorem, existence of turning point If [Equation] exists then a necessary condition that f has a turning point at c is [Equation]. Proof. Suppose [Equation], then by the second corollary to the mean value theorem (above), f is strictly increasing at c. Likewise, if [Equation] then f is strictly decreasing at c. This contradicts the supposition that f has a turning point at c; hence [Equation]. The condition is not sufficient, because it is possible that when [Equation] there is a point of inflection at c; for instance, at [Equation] for the function [Equation]. Theorem, criterion for maximum If there is a neighbourhood of c where [Equation] for [Equation] and [Equation] for [Equation], then x has a maximum at c. Proof By the second corollary to the mean value theorem (above), f is strictly decreasing for [Equation], and also strictly decreasing for [Equation]. Theorem, second derivative and maximum Suppose [Equation]. If [Equation] then f has a maximum at [Equation]. If [Equation] then f has a minimum at [Equation]. Proof This is a corollary of the preceding theorem. Suppose [Equation], then [Equation] is strictly decreasing at c. Therefore, there is a neighbourhood of c where [Equation] for [Equation] and [Equation] for [Equation], and by the preceding theorem, f has a maximum at c.
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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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