# Approximation with Taylor polynomials

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Note:% just a symbol

Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I.

Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that

f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2

and prove that limh->0 of %(h) = 1/3

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#### Solution Summary

This solution provides a proof regarding a function with a continuous third derivative.

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