1. ## derivatives

If $\displaystyle \{x_n\}$and$\displaystyle \{y_n\}$ are sequences converging to the same point $\displaystyle x_0$ such that $\displaystyle x_n$$\displaystyle <$$\displaystyle x_0$$\displaystyle <$$\displaystyle y_n$ $\displaystyle \vee$$\displaystyle n, and if \displaystyle fis differentiable funtion whose derivative is continuous at \displaystyle x_0 and which satisfies the condition that \displaystyle f(x_n)=f(y_n) \displaystyle \vee$$\displaystyle n$, show that $\displaystyle f'(x_0)=0$.

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

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

Hence $\displaystyle f'(\xi_n)=0, \forall n$ and by continuity of $\displaystyle f'$ at $\displaystyle x_0$ you conclude...

3. yes i meant$\displaystyle \forall n$ but how can you say $\displaystyle \xi_n$=$\displaystyle x_0$?

4. Originally Posted by Kat-M
yes i meant$\displaystyle \forall n$ but how can you say $\displaystyle \xi_n$=$\displaystyle x_0$?
Aw, I didn't say that. But what I can say is that $\displaystyle \lim_{n\rightarrow\infty}\xi_n=x_0$ (why? - because the sequence $\displaystyle \xi_n$ is sandwiched between two sequences that both converge to $\displaystyle x_0$). And thus, we have, by continuity of $\displaystyle f'$ at $\displaystyle x_0$:

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