Remainder Theorem

[JellyFish]

I am looking for some help in finding the Lagrange Remainder Theorem from the integral form of the remainder of a Taylor series:

R[sub:1qpkk8n8]n,a[/sub:1qpkk8n8](x) = [sup:1qpkk8n8]x[/sup:1qpkk8n8]?[sub:1qpkk8n8]a[/sub:1qpkk8n8] [f[sup:1qpkk8n8](n+1)[/sup:1qpkk8n8](t)]/n! *(x-t)dt

We are given a hint to use the mean value theorem but I'm really not sure where to start.

Thank you

 

[galactus]

Do you have to prove the Remaionder Theorem?.

Let's say h and g satisfy the Extended Mean Value Theorem on the interval [a,b]. So, there is a point c with \(\displaystyle a<c<b\) such that

\(\displaystyle \frac{h(b)-h(a)}{g(b)-g(a)}=\frac{h'(c)}{g'(c)}\)

Now, apply the Extended Mean Value Theorem over the interval [a,c].

Now, we can say there is a point d with \(\displaystyle a<d<c<b\) such that:

\(\displaystyle \frac{h'(c)-h'(a)}{g'(c)-g'(a)}=\frac{h''(d)}{g''(d)}\)

and on and on. I do nnot have time to continue right now, but see if you can proceed a little.