Question 108
Let $n \ge 2$ be an integer and $f : [a, b] \to [a, b]$ be a continuous function, differentiable on $(a, b)$, with continuous positive derivative. It is known that the equation $f'(x) = 1$ has a solution. Prove that there exists $c \in (a, b)$ such that
$$
f^{[n]}(b) - f^{[n]}(a) = (b - a) \cdot (f'(c))^{n+1},
$$
where $f^{[n]} = f \circ f \circ \dots \circ f$ (the right-hand expression has $n$ times $f$).
$$
f^{[n]}(b) - f^{[n]}(a) = (b - a) \cdot (f'(c))^{n+1},
$$
where $f^{[n]} = f \circ f \circ \dots \circ f$ (the right-hand expression has $n$ times $f$).