Page 8: Final Applications

The final page presents concluding examples and applications of function family analysis.

Example: Analysis of F(x) = 1/3x³ - ax, demonstrating:

1. Derivative calculations
2. Critical point determination
3. Second derivative analysis

Highlight: Integration of all previously learned concepts in comprehensive examples.

Page 2: Extrema Analysis

This page delves into the calculation of extreme points for function families, demonstrating the systematic approach to finding maxima and minima.

Definition: Extreme points are locations where a function reaches its maximum or minimum values.

Example: For fa(x) = x² - 2ax + 8a - 16, the process involves:

1. Finding the first derivative (f'a(x) = 2x - 2a)
2. Finding the second derivative (f"a(x) = 2)
3. Analyzing necessary and sufficient conditions

Highlight: The extremstellen berechnung mathematische funktionen process requires both necessary and sufficient conditions to be checked.

Page 4: Practical Applications

This section presents practical examples of function analysis, focusing on slope conditions and parameter determination.

Example: Analysis of fa(x) = -2x + a with conditions:

1. Value at x = 0
2. Slope = 1 at specific points
3. Parameter determination

Highlight: Special attention is given to definition domains and parameter restrictions.

Page 6: Complex Function Analysis

This section deals with more sophisticated function family analysis, including quadratic and higher-order terms.

Example: Analysis of fa(x) = x² - ax + 4, including:

1. Derivative calculation
2. Critical point determination
3. Parameter value analysis