ignore this
Suppose is such that both and exist for all , so that Taylor's theorem tells us that, for each there is a such that so that . Suppose further that on [0,2] the inequalities and hold. Write down the Taylor expansions of f(0) and f(2) about the point , using the above form of Taylor's Theorem, with a remainder involving . Hence prove that for all we have .
I'm not sure about the first part, what I have is:
by setting a+h=0 and
but I don't think this is correct. It says about the point x in [0,2] but I don't know how to apply this to the formula. I have no idea for the last part. Thanks!
This is an infuriating problem. I have come across variants of it many times, and it always takes me ages to get the answer to come out right. You need to do the steps of the argument in exactly the right order.
From the first equation,
.
From the second equation,
.
Add those equations:
.
Therefore
.
But , and
.
Thus .