A Perfect Set For JEE!

$f'(x)$ 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 $f'(k) = k$ .