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

• Apr 8th 2010, 11:06 PM
alice8675309
numerical analysis? show that if f is in c^2[a,b]...
Show that if f $\in$ $C^2$[a,b], then for $x_1$, $x_2$ $\in$[a,b], there exists $\xi$ $\in$[a,b], so that f '( $x_1$)=(f( $x_2$)-f( $x_1$)/( $x_2$- $x_1$) )-(( $x_2$- $x_1$)/2) f''( $eta$)

To do this, do I have to show the proof? If so what is the proof?
• Apr 10th 2010, 11:56 PM
CaptainBlack
Quote:

Originally Posted by alice8675309
Show that if f $\in$ $C^2$[a,b], then for $x_1$, $x_2$ $\in$[a,b], there exists $\xi$ $\in$[a,b], so that f '( $x_1$)=(f( $x_2$)-f( $x_1$)/( $x_2$- $x_1$) )-(( $x_2$- $x_1$)/2) f''( $eta$)

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

Write out the Taylor expansion of $f(x)$ about $x_1$ evaluated at $x_2$, 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 $[x_1,x_2]$ if $x_2>x_1$ and $[x_2,x_1]$ if $x_2.

Now rearrange into the form:

$f'(x_1)= ...$

CB
• Apr 24th 2010, 12:28 AM
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