# A Perfect Set For JEE!

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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