Graph Shape Linked to the Signs of the First Two Derivatives

This example demonstrates how the overall shape of a function’s graph is determined by the signs of its first and second derivatives: the sign of the first derivative indicates increasing/decreasing behavior, while the sign of the second derivative indicates concavity (and helps explain how curvature changes near inflection points). Key concepts include first and second derivatives, tangent-line slope, monotonicity, concavity/convexity, and the qualitative interpretation of derivative signs on the graph. The construction is interactive and uses GeoGebra objects to display both the function and derivative information: Vector and Textfield are used for vector/tangent-style annotations and explanatory labels, while toggling/display controls such as pokazfunkcje, pokazpochodna, and pokazwykresdrugiejpochodnej show the function, the first derivative, and the second-derivative graph/markers. InputBox1 and multiple buttons plus text objects (e.g., tekst8/tekst9 and other text fields) organize the visualization and user interaction. Suitable for high school and early undergraduate calculus lessons, especially for building intuition between derivative signs and graph behavior.