What is an intuitive explanation for the fact that the remainder of a Taylor polynomial has this equivalency? Why is z necessarily between x and c? Why does such a value for z exist in the first place?
closed as off-topic by Andrés E. Caicedo, user10354138, John B, max_zorn, Gibbs Nov 9 '18 at 22:06
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For n = 0, it's applying the mean value theorem to f(x).
For n = 1, it's applying the mean value theorem to
f(c) + f'(c)(x - c).
And so to n. So it's basically an elaborate MVT.