Problem $ f(x)$ is defined over $ [a,b]$ and differentiable over $ (a,b)$ , where $ b-a\geq 4.$ Prove that there exists $ \xi \in (a,b)$ such that $ f'(\xi)<1+f^2(\xi)$ . My Proof Since $ b-a \geq 4$ ，we can obtain $ $ \exists x_1,x_2 \in (a,b):x_2-x_1>\pi.$ $ Denote$ $ F(x):=\arctan f(x).$ $ Obviously, $Read more