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We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter.

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Let f:(-1,1) --> R, an odd function [f(-x) = -f(x)],
five times differentiable. Prove that for all
x in (-1,1), there exists theta (dependent on x) in (0,1) such that:

f(x) = (1/3)(x)[f '(x)+2f '(0)]-(1/180)(x^5)(f'''''(theta(dependent on x)x))

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- theta(dependent on x) is symbol of theta subscript x
- f''''' is the fifth derivative

Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. The same problem is also attached as a word document with all the symbols.