Question: Suppose that n N is odd and f(n) exists on [a, b]. If f(k)(a) = f(k)(b) = 0 for all k = 0, 1,

Suppose that n ∈ N is odd and f(n) exists on [a, b]. If f(k)(a) = f(k)(b) = 0 for all k = 0, 1, ...,n - 1 and f(c) ≠ 0 for some c ∈ (a, b), prove that there exist x1, x2 ∈ (a, b) such that f(n)(x1) is positive and f(n)(x2) is negative.

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