Rolle’s Theorem Visual Construction and Tangent Point

This GeoGebra example demonstrates Rolle’s Theorem by constructing a differentiable function on an interval where the endpoint values are equal. Visually, it highlights that at least one interior point exists where the derivative equals zero—equivalently, where the graph has a horizontal tangent. The key mathematical ideas are Rolle’s Theorem (existence of a stationary point given equal endpoint values and differentiability), the interpretation of f′(x)=0 as zero slope of the tangent, and the use of geometric constraints to locate the tangent/critical point. In the construction, it uses commands such as Line (straight lines), Intersect (intersection points), Roots (finding roots/zero-value positions), Tangent (tangent lines), Circle (auxiliary geometric constraints/objects), Segment (segments), along with Min/Max for extrema and labeled objects for the critical points and intermediate intersections (e.g., f, tangent-related points like t, and markers such as h/h′, l/m, and multiple intersection/parameter points). Suitable for upper-secondary enrichment, first-year calculus instruction, and classroom demonstrations of derivative concepts and theorem verification.