Explain Rolle’s Theorem with a sketch.

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1. Rolle's Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.
2. To apply Rolle's Theorem, first ensure that the function meets the criteria: check for continuity on [a, b] and differentiability on (a, b).
3. Next, verify that the function values at the endpoints are equal, i.e., f(a) = f(b). This is crucial for the theorem to hold.
4. Once the conditions are satisfied, find the derivative f'(x) of the function and set it equal to zero to solve for c, which represents the point(s) where the tangent to the curve is horizontal.
5. Finally, sketch the function on the interval [a, b], marking the points (a, f(a)) and (b, f(b)) at the same height, and indicate the point c where the derivative is zero, showing that the slope of the tangent line at that point is horizontal.