I have a question concerning the Mean Value Theorem (and maybe Rolle's Theorem).
In my calc book by Stewart, the concept of both theorems seemed to be thrown out of nowhere with a bunch of conditions and statements like.
"If $f$ is differentiable on an open interval $(a,b)$, then..."
So here is what i don't understand, why can't the interval be closed for differentiability? And what was the motivation behind creating the two theorems? I don't see how anyone, one day, could sit down and just write down a bunch of rules and conclude a formula and give a name to it.
I won't take the equation as face value and accept it like the rest of the mindless sheeps in my university