Difference between revisions of "Lagrange's Mean Value Theorem"
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| − | Let <math>f: | + | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> |
Let <math>f</math> be continous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>. | Let <math>f</math> be continous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>. | ||
Revision as of 10:25, 15 February 2008
Lagrange's mean value theorem or LMVT is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using LMVT.
Statement
Let ![$f:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/6/7/c/67c320754a36aaec178e61f04bfcbac41c0bf675.png)
Let 
be continous on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
and differentiable on 
.
Then 
such that 
Proof
We reduce the problem to the Rolle's theorem by using an 'auxillary function'.
Consider 
note that 
By Rolle's theorem, 
such that 
i.e. 
or
QED
