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Calculus Level 5

\(f'(x) \) maps from \( [0,1] \to [ p(a), p(b) ] \). Given that p is a differentiable function on [a,b] and \(p(g(x)) = x\), \(a=g(0)\) and \(b = g(1) \). Which of the following is/are true?

(A): \(f(0) +2 < f(1) \).

(B): \( f(1) \leq 1 + f(0) \).

(C): \( \dfrac{\int_0^1 f'(x) \, dx}{\int_0^1 g'(x) \, dx } \leq p'(c) \). for some \(c \in (a,b) \).

(D): There exists a \(k\in [0,1] \) such that \(f'(k) = k\).

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