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

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Apr 8th 2010, 10:06 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, 10:56 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 $x_2<x_1$ .

Now rearrange into the form:

$f'(x_1)= ...$

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

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