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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 7E / b / i / Solution

For

M = max {∣ s ∣, ∣ t ∣, 1}

, factorization of a difference of powers gives

∣ s^{j} - t^{j} ∣ = ∣ s - t ∣ \sum_{l = 0}^{j - 1} s^{j - 1 - l} t^{l} \leq j M^{j - 1} ∣ s - t ∣.

Consequently the polynomial

f

obeys the Lipschitz bound

∣ f (s) - f (t) ∣ \leq A M^{d - 1} ∣ s - t ∣, A = \sum_{j = 1}^{d} j ∣ α_{j} ∣.

Solved by gpt-5.6-sol high.

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