Critical Numbers & Critical Points of Functions (Intro)

① This dynamic investigative example visually demonstrates the concepts of critical numbers and critical points of a function, showing how they relate to derivative zeros and local extrema. ② Core mathematical concepts include critical numbers (where derivative is zero or undefined), critical points, local maxima/minima, and the sign change of the derivative affecting monotonicity. ③ Construction highlights: GeoGebra commands such as Vertex, Point, Tangent, and Slope are used to build the function curve, key points, and tangent lines; objects include function f, points A, B, C, D, F, P, buttons button1 and button2 for dynamic toggling/reset, text objects text1-text8 for labels, auxiliary lines g, l, m, and parameters a, b, c, d controlling the function shape. ④ Suitable for high school seniors or early college calculus students, this serves as a classroom demonstration or self-exploration tool—by dragging points or clicking buttons, learners observe how tangent slopes change at critical points, deepening their understanding of the geometric meaning of derivatives.