Question 86
Let $f: [0, 1] \to [0, 1]$ be an increasing and derivable function that has an inverse $f^{-1}$.
If $\int_0^1 f(x) dx = \int_0^1 f^{-1}(x) dx$, prove that there exist two distinct real numbers $a$ and $b$, $0 \le a \lt b \le 1$, such that $f'(a) = f'(b) = 1$.
If $\int_0^1 f(x) dx = \int_0^1 f^{-1}(x) dx$, prove that there exist two distinct real numbers $a$ and $b$, $0 \le a \lt b \le 1$, such that $f'(a) = f'(b) = 1$.