Use the Mean Value Theorem to prove that for all a,b that belongs to the reel, we have :

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- January 26th 2009, 03:39 PMLarriotoMean Value Theorem Proof
Use the Mean Value Theorem to prove that for all a,b that belongs to the reel, we have :

- January 26th 2009, 04:03 PMNonCommAlg
- January 26th 2009, 04:15 PMLarrioto
There is a second part to this problem :

Assume that f : is a differentiable function and that there exists a real number M > 0 such that |f '(x)| for all x in the reel then show that :

for all x in the reel - January 26th 2009, 04:25 PMMathstud28
Choose any value , we know there exists a corresponding interval such that and since was arbitrary this must work for every interval .

Another almost identical method would be to assume that there exists an interval such that . But since is differentiable we can find a such that which is a contradiction.

This is called Lipschitz's continuity by the way.

If you have any questions at all feel free to ask :) - January 26th 2009, 04:38 PMLarrioto
- January 26th 2009, 04:59 PMMathstud28
- January 26th 2009, 05:09 PMLarrioto
ok , i get it , it's just that f(c) can be replaced by f(x), that was confusing me ...

- January 26th 2009, 05:46 PMNonCommAlg
this, although an interesting claim, but is not what mean value theorem says: you choose an interval [a,b] and then you find a < c < b that satisfies the condition. what you're doing here is

reversing things, i.e. you choose c and then you claim that there exists an interval [a,b] containing c with that condition! i'm not sure if this is always possible! - January 26th 2009, 06:10 PMMathstud28
- January 26th 2009, 07:15 PMNonCommAlg
here's a simple counter-example: let and choose geometrically it's clear that there are no such that you can also prove it very easily:

if there exist such then: which is possible iff because:

regarding**Larrioto**'s question:

by mean value theorem for any real numbers there exists (between ) such that thus: