Suppose that f : [a, b] → [c, d] is differentiable and onto. If f ′ is never zero on [a, b] and d − c ≥ 2, prove that for every x ∈ [c, d] there exist x1 ∈ [a, b] and x2 ∈ [c, d] such that | f ′(x1)(f −1(x) − f −1(x2))| = 1.
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