numerical analysis? show that if f is in c^2[a,b]...

• Apr 8th 2010, 10:06 PM
alice8675309
numerical analysis? show that if f is in c^2[a,b]...
Show that if f$\displaystyle \in$$\displaystyle C^2[a,b], then for \displaystyle x_1,\displaystyle x_2$$\displaystyle \in$[a,b], there exists $\displaystyle \xi$$\displaystyle \in[a,b], so that f '(\displaystyle x_1)=(f(\displaystyle x_2)-f(\displaystyle x_1)/(\displaystyle x_2-\displaystyle x_1) )-((\displaystyle x_2-\displaystyle x_1)/2) f''(\displaystyle eta) To do this, do I have to show the proof? If so what is the proof? • Apr 10th 2010, 10:56 PM CaptainBlack Quote: Originally Posted by alice8675309 Show that if f\displaystyle \in$$\displaystyle C^2$[a,b], then for $\displaystyle x_1$,$\displaystyle x_2$$\displaystyle \in[a,b], there exists \displaystyle \xi$$\displaystyle \in$[a,b], so that f '($\displaystyle x_1$)=(f($\displaystyle x_2$)-f($\displaystyle x_1$)/($\displaystyle x_2$-$\displaystyle x_1$) )-(($\displaystyle x_2$-$\displaystyle x_1$)/2) f''($\displaystyle eta$)

To do this, do I have to show the proof? If so what is the proof?

Write out the Taylor expansion of $\displaystyle f(x)$ about $\displaystyle x_1$ evaluated at $\displaystyle x_2$, with remainder term for the series truncated after the first linear term (this will be in terms of the second derivative at some point in the interval $\displaystyle [x_1,x_2]$ if $\displaystyle x_2>x_1$ and $\displaystyle [x_2,x_1]$ if $\displaystyle x_2<x_1$.

Now rearrange into the form:

$\displaystyle f'(x_1)= ...$

CB
• Apr 23rd 2010, 11:28 PM
CaptainBlack
Quote:

Originally Posted by nike22
I also want to know the answer to this question.

And what is it about the suggested approach that you don't understand?

CB