Let $f(x)$ be a twice differentiable function over $[a,b]$ with arc length $L$. Show that there exists a value $c \in [a,b]$ such that the angle $\theta$ between the tangent line and the horizontal line at $x = c$ is given by $\cos\theta = \frac{b-a}{L}.$
If you draw this out, you can observe that the distance of the tangent line on $[a,b]$ must be equal to the arc length $L$, but I have no idea how to prove that. Any help is appreciated!