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Necessary values of f'-f

Source: 2023 IMC P7

August 3, 2023
calculusIMCfunctionIMC 2023

Problem Statement

Let VV be the set of all continuous functions f ⁣:[0,1]→Rf\colon [0,1]\to \mathbb{R}, differentiable on (0,1)(0,1), with the property that f(0)=0f(0)=0 and f(1)=1f(1)=1. Determine all α∈R\alpha \in \mathbb{R} such that for every f∈Vf\in V, there exists some ξ∈(0,1)\xi \in (0,1) such that f(ξ)+α=f′(ξ)f(\xi)+\alpha = f'(\xi)
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