**Rolle's Theorem:**

Rolle's Theorem is a fundamental result in calculus that provides a specific condition under which a function must have a horizontal tangent (i.e., a point where the derivative is zero) within a given interval.

**Statement:**

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

1. **Continuity**: \( f \) is continuous on the closed interval \([a, b]\).
2. **Differentiability**: \( f \) is differentiable on the open interval \( (a, b) \).
3. **Equal Endpoints**: \( f(a) = f(b) \).

Then, there exists at least one point \( c \) in the open interval \( (a, b) \) such that:

\[
f'(c) = 0
\]

**Proof:**

To prove Rolle's Theorem, we proceed as follows:

1. **Consider the Case \( f(x) \) is Constant:**

   If \( f(x) \) is constant on \([a, b]\), then \( f(x) = f(a) = f(b) \) for all \( x \) in \([a, b]\). In this case, the derivative \( f'(x) = 0 \) for all \( x \) in \((a, b)\). Hence, any \( c \in (a, b) \) satisfies \( f'(c) = 0 \).

2. **Consider the Case \( f(x) \) is Not Constant:**

   Since \( f(x) \) is continuous on \([a, b]\) and differentiable on \( (a, b) \), the Extreme Value Theorem guarantees that \( f(x) \) attains its maximum and minimum values at some points in \([a, b]\).

   Let \( x_0 \) be a point in \([a, b]\) where \( f(x) \) attains its maximum value, and let \( x_1 \) be a point where \( f(x) \) attains its minimum value. Since \( f(a) = f(b) \), both \( x_0 \) and \( x_

Understand Rolle's Theorem in calculus, which states that for a continuous and differentiable function on a closed interval [a, b], if f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0. Learn the proof and implications of Rolle's Theorem with detailed explanations and examples. Suitable for college-level students and calculus enthusiasts.

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