Mean Value Theorems
DOWNLOAD
FREE
|
Cauchy’s mean value theorem
Equations are omitted for technical reasons - download the original pdf
If the functions [Equation goes here - download the original to see it.] and [Equation] have differential coefficients which do not vanish simultaneously in the interval a, b, and if [Equation] is not equal to [Equation], then there is a number [Equation] between a and b for which [Equation] Proof Let [Equation] be the function defined by [Equation] where A and B are constants chosen so that [Equation] [Equation]
This is a set of linearly independent equations that can be solved uniquely for A and B since [Equation]. By Rolle’s theorem, there is a number [Equation] between a and b such that [Equation]. That is, by differentiating (1) [Equation] Now [Equation] since if it were this equation would make [Equation] also, which is contrary to the first hypothesis. Hence we may divide by [Equation] to obtain [Equation] Now [Equation] Thus, since [Equation] and [Equation] [Equation] Hence [Equation]
|
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
|
