Suppose that exists on and and p is the polynomail of degree such that . I want to show that for :
for
We are just starting Taylor's Theorem but I'm not sure how this works. Any advice?
The only information I have about f(x) is given above. I was trying to introduce a new polynomial that did to try to get the rest of the right hand side. But I kept getting stuck because I don't know anything about f(x) or p(x). I knew I could say:
But all I know about p(x) is that so I thought if I subtracted the 2 in the new polynomial, I might get close because of this equality... but it doesn't end up giving me anything that looks like what I want to show because I don't know about the derivatives of p...
I just realized theres a hint in the back of the book that says to do the following:
let x be fixed and distinct from and consider function:
where we choose K so that
then use Rolles thm to show for
Rolles???? How does that relate at all? this confused me more than helped i think? any ideas?