The usual formulation of the mean value theorem in a real analysis course is something like this:
Let $f\colon [a,b] \to \mathbb{R}$ be continous on $[a,b]$ and differentiable on $]a,b[$. Then there is a $\xi \in ]a,b[$ such that
$$ f(b) - f(a) = f'(\xi)(b-a) $$
Since differentiability implies continuity one could impose the slightly less general condition that $f$ should be just be differentiable on $[a,b]$.
Are there any "non exotic" cases or any theorems with are proven using the mean value theorem where one really need the more general form above?
Is it correct that I need only the less general version for deriving the following standard calculus theorems:
- characterizing monotonic and constant function with the derivative
- the inverse function theorem
- the second derivative test for minima/maxima and the change of sign test