Prove Rolles Theorem

Prove Rolle's Theorem: Suppose $f$ is continuous on a closed interval $[a, b]$ and differentiable on the corresponding open interval $(a, b)$. Show that if $f(a) = f(b)$, then there exists a $c$ in $(a, b)$ such that $f'(c) = 0$.

Rolle's Theorem is a fundamental result in real analysis that provides a clear connection between the values of a continuous function on a closed interval and the behavior of its derivative. At a high level, this theorem highlights the idea that a differentiable function which starts and ends at the same value over a closed interval must have a point where its instantaneous rate of change, or derivative, is zero. This effectively means that the graph of the function has at least one horizontal tangent line in the interval. Therefore, the crux of proving Rolle's Theorem lies in leveraging the conditions of continuity and differentiability along with the fact that the initial and final points of the interval share the same functional value.

The proof typically utilizes key analytical concepts such as the Intermediate Value Theorem and the properties of derivatives, helping establish the presence of a critical point in the function's journey across the interval. This theorem is foundational as it lays the groundwork for more advanced results such as the Mean Value Theorem, which generalizes the concept further by not requiring equal endpoints but instead relates the average rate of change over an interval to an instantaneous rate of change at some point within the interval. Understanding Rolle's Theorem thus not only solidifies one's grasp of fundamental differentiability properties but also prepares the stage for broader applications across calculus and real analysis.