For background, the Mean Value Theorem says that if $f\in C^1([a,b])$, then there exists $c\in(a,b)$ such that $$f'(c) = \frac{f(b) - f(a)}{b-a}.$$
My question is, how does $c$ vary as a function of properties of $f$, $a$, and $b$?
For some $f$, we have an easy closed form for $c$. Take $f(x) = x^3$, $a = 0$, and $b$ an arbitrary $x$ (wlog, assume $x > 0$), for example. Then we have the equation
$$\frac{x^3}{x} = 3c^2,$$
so that $c = \frac{x}{\sqrt{3}}$.
But what if the function is more complicated? Obviously we could just solve numerically, but is there a theorem that can tell us anything interesting about $c$? For example, a theorem that tells us when $c$ is a polynomial function of $x$? Or an algebraic function? Or how close to $x / 2$ we can expect $c$ to be? How good an estimate $c = x/2$ is? How higher derivatives affect whether $c$ is closer to one or the other endpoint? Etc.
Running list of insights
- We may assume $a = 0$ and $f(0) = 0$ without loss of generality because derivatives are well-defined up to addition by a constant and phase shifts do not affect the underlying problem.