Page 1: Introduction and Initial Calculations

This page introduces the main problem: analyzing a cubic function f(x) = x³ - 5x² - x + 12.

Key calculations:

1. Derivatives up to the third order
2. Y-intercept
3. Roots of the function

Definition: A ganzrationale Funktion 3. grades (cubic function) is a polynomial function of degree 3, typically in the form f(x) = ax³ + bx² + cx + d.

Example: The given function f(x) = x³ - 5x² - x + 12 is a cubic function.

The page demonstrates how to find roots using polynomial division and provides a step-by-step approach to solving the problem.

Highlight: The use of polynomial division to find roots is a crucial technique for solving higher-degree polynomial equations.

Page 4: Extrema and Inflection Point Analysis

This page focuses on analyzing extrema and finding the inflection point of the ganzrationale Funktion 3. grades.

Key calculations:

1. Determining coordinates of extrema
2. Proving whether a point is a maximum or minimum
3. Calculating the inflection point

Example: The maximum point is found to be at (0.19, 12.1011), while the minimum is at (3.52, -0.69).

The inflection point is calculated using the second derivative and verified using the third derivative.

Highlight: The process of verifying extrema and inflection points demonstrates the importance of higher-order derivatives in curve analysis.