Difference between revisions of "Lagrange's Mean Value Theorem"
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| − | '''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. | + | '''Lagrange's mean value theorem''' (often called "the mean value theorem," and abbrviated MVT 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== | ==Statement== | ||
| − | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> | + | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> be a [[continuous function]], [[differentiable]] on the [[open interval]] <math>(a,b)</math>. Then there is some <math>c\in (a,b)</math> such that <math>f'(c)=\frac{f(b)-f(a)}{b-a}</math>. |
| − | + | Informally, this says that a differentiable function must at some point grow with instantaneous velocity equal to its average velocity over an interval. | |
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==Proof== | ==Proof== | ||
Revision as of 16:41, 15 February 2008
Lagrange's mean value theorem (often called "the mean value theorem," and abbrviated MVT 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)
be a continuous function, differentiable on the open interval 
. Then there is some 
such that 
.
Informally, this says that a differentiable function must at some point grow with instantaneous velocity equal to its average velocity over an interval.
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
