Rolle's theorem
Rolle's theorem
Rolle's theorem states that if a real-valued function \[f\] is continuous on a interval \[\left[ a,b \right]\], differentiable on the interval \[\left( a,b \right)\], and \[f(a)=f(b)\], then there exists one \[c\] in the interval \[\left( a,b \right)\] such that \[f^{\prime}(c)=0\].
Essentially, the theorem tells us that any function have two (or more) points that share the same value is guaranteed to have some point where it's derivative is zero (stationary point).
Also, the reason why both open and closed intervals are used in the definition is that the extreme value portion of the proof requires continuity on a closed interval while at the endpoints \[a\] and \[b\], we do not require \[f\] to be differentiable because derivatives are typically one-sided there (or not defined if extended beyond the domain).
Proof
The idea of the proof is to argue that if \[f(a)=f(b)\], then \[f\] must attain either a maximum or minimum, call it \[c\], somewhere between \[a\] and \[b\]. In particular, if the derivative exists at \[c\], it must be zero.
By assumption, \[f\] is continuous on \[\left[ a,b \right]\], and by the extreme value theorem it attains both the maximum and minimum in \[\left[ a,b \right]\].
If the endpoints of \[\left[ a,b \right]\] are both the maximum and minimum (which means the function is a constant), then the derivative of \[f\] at every point in \[\left( a,b \right)\] must also be zero.
Now suppose that the maximum is obtained at some \[c\in \left( a,b \right)\] (the argument for a minimum is identical). Since \[c\] is a point where \[f\] attains a local extremum (maximum/minimum), we can apply the interior extremum theorem, which tells us that the derivative of a local extremum is zero, thereby proving that the derivative at \[c\] is zero.