Rolle’s Theorem:

If a function $f(x)$ satisfies the following three conditions on a closed interval $[a, b]$:

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

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

This means the function has at least one point where the slope of the tangent is zero (a horizontal tangent) within the interval.

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Related Questions:

1. What is the geometrical interpretation of Rolle’s Theorem?
2. Can Rolle’s Theorem apply to a function that is not differentiable everywhere?
3. How does Rolle’s Theorem relate to the Mean Value Theorem?
4. Provide an example function that satisfies Rolle’s Theorem on a given interval.
5. How can one verify if a function meets the conditions of Rolle’s Theorem?

Tip:

Always ensure to check both continuity and differentiability conditions before applying Rolle’s Theorem; they are key to its validity.