Rolle's theorem as stated:

 

let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a)=f(b) then there is at least one number c in (a,b) such that f'©=0.

 

Let's say f(a) does not equal f(b), but there are two other points d and e within the interval (a,b) such that f(d)=f(e). Can Rolle's Theorem be used/defined that way?

 

This is a calc 1 class, so if you could keep your answers within that realm, i'd appreciate it. Thanks!

Rolle's theorem can still be applied, but you'd consider it on the function f:[d,e]->R (assuming of course that d < e).