Page 1: Advanced Derivative Analysis

This page from the Lambacher Schweizer Kursstufe - Leistungsfach Lösungen covers solutions to problem 10 on page 28, demonstrating advanced techniques in calculus for analyzing polynomial functions and their derivatives.

The solutions are divided into two parts:

a) Analysis of f(x) = x³ - ax²

The solution begins by finding the first, second, and third derivatives of the given function. It then solves for critical points by setting f'(x) = 0 and determines the nature of these points using the second derivative test.

Example: f'(x) = 3x² - 2ax, f''(x) = 6x - 2a, f'''(x) = 6

The solution also considers the parameter 'a' and its effect on the function's behavior.

b) Analysis of f(x) = x⁴ - 2ax² + 1

This part follows a similar approach but with a more complex function. It calculates derivatives up to the fourth order and analyzes critical points.

Highlight: The solution demonstrates how to find the x-coordinate of the inflection point using the second derivative.

Vocabulary: Inflection point - A point on a curve at which the curvature changes from convex to concave or vice versa.

The page concludes with evaluating the function and its derivatives at specific points, showcasing the application of the chain rule and substitution techniques.

Definition: Chain Rule - A formula for computing the derivative of a composite function.

This comprehensive solution exemplifies the depth of analysis expected in the Lambacher Schweizer Mathematik für Gymnasien curriculum, providing students with a thorough understanding of derivative applications in function analysis.