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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