Assignment:
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
Provide complete and step by step solution for the question and show calculations and use formulas.