Let be a real-valued continuously differentiable function on $[a, b]$ with positive derivative on $(a, b)$. Prove that, for all pairs with and , there exists such that
Because , and are the same sign. Without lost generality, we can assume that and are positive (Otherwise, we consider -f). Then for all , by Lagrange’s Theorem we get , hence .
Now, consider two function: and are continuously differentiable on , by Cauchy’s Theorem there is