Let [a, b] be a closed, bounded, nondegenerate interval. Find all functions f which satisfy the following

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Let [a, b] be a closed, bounded, nondegenerate interval. Find all functions f which satisfy the following conditions for some fixed a > 0 : f is continuous and 1-1 on [a, b], fʹ(x) ≠ 0 and f'(x) = α(f-l)'(f(x)) for all x ∈ (a, b).
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