Prove: if is differentiable on and , then .

How do I approach such an exercise? Any ideas would be appreciated.

Printable View

- Jul 12th 2012, 03:27 AMloui1410Fermat, Rolle and Lagrange
Prove: if is differentiable on and , then .

How do I approach such an exercise? Any ideas would be appreciated. - Jul 12th 2012, 05:08 AMthesmurfmasterRe: Fermat, Rolle and Lagrange
You know that for any you can find an so that . In particular, this says that for any you can find an x so that . Can you see why that is? Can you then conclude anything about your limit?

- Jul 12th 2012, 05:24 AMloui1410Re: Fermat, Rolle and Lagrange
- Jul 12th 2012, 10:21 AMthesmurfmasterRe: Fermat, Rolle and Lagrange
Let's say that for all (we know by definition that we can find such an N for each ). From to we're changing the input by x (2x-x=x). Since the absolute value of the derivative is always less than epsilon in this interval, we can definately not move further from f(x) than when we move x units ahead. Therefore .

- Jul 12th 2012, 10:29 AMthesmurfmasterRe: Fermat, Rolle and Lagrange
Let's say that for all (we know by definition that we can find such an N for each ). From to we're changing the input by x (2x-x=x). Since the absolute value of the derivative is always less than epsilon in this interval, we can definately not move further from f(x) than when we move x units ahead. Therefore .

- Jul 12th 2012, 10:44 AMloui1410Re: Fermat, Rolle and Lagrange