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  1. #1
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    derivatives

    If \{x_n\}and \{y_n\} are sequences converging to the same point x_0 such that x_n < x_0 < y_n \vee n, and if fis differentiable funtion whose derivative is continuous at x_0 and which satisfies the condition that f(x_n)=f(y_n) \vee n, show that f'(x_0)=0.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    If \{x_n\}and \{y_n\} are sequences converging to the same point x_0 such that x_n < x_0 < y_n \vee n, and if fis differentiable funtion whose derivative is continuous at x_0 and which satisfies the condition that f(x_n)=f(y_n) \vee n, show that f'(x_0)=0.
    Does x_n<x_0<y_n\vee n actually mean x_n<x_0<y_n, \forall n? If so, I would advise you to use the mean value theorem that guarantees \forall n the existence of a \xi_n such that x_n <\xi_n<y_n and

    f'(\xi_n)\cdot(y_n-x_n)=f(y_n)-f(x_n)=0

    Hence f'(\xi_n)=0, \forall n and by continuity of f' at x_0 you conclude...
    Last edited by Failure; August 7th 2009 at 08:56 AM.
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  3. #3
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    yes i meant \forall n but how can you say \xi_n= x_0?
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by Kat-M View Post
    yes i meant \forall n but how can you say \xi_n= x_0?
    Aw, I didn't say that. But what I can say is that \lim_{n\rightarrow\infty}\xi_n=x_0 (why? - because the sequence \xi_n is sandwiched between two sequences that both converge to x_0). And thus, we have, by continuity of f' at x_0:

    f'(x_0)=f'(\lim_{n\rightarrow \infty}\xi_n)=\lim_{n\rightarrow\infty}f'(\xi_n)=\  lim_{n\rightarrow\infty}0=0
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