Explore Critical Points/Numbers: Interactive Derivative-Based Visualization

① This GeoGebra applet demonstrates how to identify critical points of a function based on the formal definition: points where the derivative is zero or does not exist, with dynamic visual feedback on the graph. ② Core mathematical concepts include the definition of critical numbers, the geometric meaning of the derivative (tangent slope), and the relationship between derivative sign changes and potential local extrema. ③ Construction highlights: the Derivative command generates f', Textfield and InputBox1 allow users to input the function and candidate x-values, the Defined command checks whether the derivative exists at a point, and objects such as text1, text2, text3, a, b, c, d, e, g, A, B, and pic1 are used to display explanatory text, mark points, draw tangent lines, define the domain, and set the background image. ④ Suitable for high school or introductory college calculus students, this applet serves as an interactive tool for classroom demonstration or self-study, deepening understanding of critical points, derivative existence, and the first-derivative test.