Rolle's Theorem is a fundamental result in calculus, which states that if a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one point $c$ in $(a, b)$ such that the derivative of the function at $c$ is zero, i.e., $f'(c) = 0$.

Statement of Rolle's Theorem:

Let $f: [a, b] \rightarrow \mathbb{R}$ be a function that satisfies the following conditions:

1. $f$ is continuous on the closed interval $[a, b]$,
2. $f$ is differentiable on the open interval $(a, b)$,
3. $f(a) = f(b)$.

Then, there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.

Proof of Rolle's Theorem:

1. Existence of a Maximum or Minimum:

   Since $f$ is continuous on the closed interval $[a, b]$, by the Extreme Value Theorem, $f$ must attain its maximum and minimum values on $[a, b]$. Let $M$ and $m$ be the maximum and minimum values of $f$ on $[a, b]$ respectively. Therefore, there exist points $x_1, x_2 \in [a, b]$ such that: $f(x_1) = M \quad \text{and} \quad f(x_2) = m.$

2. Case 1: Maximum or Minimum at an Interior Point

   If either $x_1$ or $x_2$ lies in the open interval $(a, b)$, say $x_1 \in (a, b)$, then $f$ attains a local maximum at $x_1$. Since $f$ is differentiable at $x_1$, by Fermat's Theorem, the derivative at $x_1$ must be zero: $f'(x_1) = 0.$ Similarly, if $x_2 \in (a, b)$, then $f'(x_2) = 0$.

3. Case 2: Maximum or Minimum at the Endpoints

   If both $x_1 = a$ and $x_2 = b$, then $f(a) = f(b)$. By assumption $f(a) = f(b)$, meaning $f(a) = f(b) = M = m$. This implies that the function $f$ is constant on $[a, b]$. For a constant function, the derivative is zero everywhere on $(a, b)$. Hence, for every $c \in (a, b)$, $f'(c) = 0$.

4. Conclusion:

   In both cases, there exists at least one point $c \in (a, b)$ such that $f'(c) = 0$.

This concludes the proof of Rolle's Theorem.

---

Would you like to see the theorem applied in a specific example, or do you have any questions?

Further Exploration:

1. What are the practical applications of Rolle's theorem?
2. Can Rolle's theorem be extended to functions of several variables?
3. How does Rolle's theorem relate to the Mean Value Theorem?
4. What are the differences between Rolle's theorem and the Mean Value Theorem?
5. How can Rolle's theorem be visualized geometrically?
6. Under what conditions does Rolle's theorem fail?
7. How does differentiability influence the conclusions drawn from Rolle's theorem?
8. Can you provide an example where Rolle's theorem does not apply?

Tip: Rolle's Theorem is often used as a step in proving other important results in calculus, like the Mean Value Theorem. Understanding it thoroughly can help in grasping more complex concepts.