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!
From the first equation,
From the second equation,
Add those equations:
But , and