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
Best Answer
The intuitive motivation for Rolle’s theorem is pretty simple. If $f(a)=0=f(b)$, then either $f$ is constant, or its value moves away from and then back to $0$. If $f$ is continuous, its value can’t jump instantaneously back to $0$: it has to turn around. If in addition $f$ has a derivative at every point between $a$ and $b$, its value can’t make a sharp turn: it has to turn gradually. And if it turns gradually, intuition says that at some point during the turn it has to be ‘moving’ horizontally $-$ which is exactly what Rolle’s theorem says in a rigorous way.
This also explains (at the intuitive level) why we don’t care about differentiability at the endpoint: the turnaround has to be somewhere in $(a,b)$, not at an endpoint of the interval.
The intuition behind the Mean Value Theorem is pretty much the same, except that horizontal is replaced by in the direction from $\langle a,f(a)\rangle$ to $\langle b,f(b)\rangle$.