Calculating Extrempunkte and Wendepunkte Using Derivatives

This page provides a comprehensive guide on how to calculate Extrempunkte (extreme points) and Wendepunkte (inflection points) using derivatives. The process is demonstrated through a detailed example.

For the given function f$x$ = 3x³ - 6x² + 2, the page outlines the steps to find both types of critical points.

Definition: Extrempunkte are points where a function reaches its maximum or minimum values, while Wendepunkte are points where the curvature of a function changes.

To find Extrempunkte, the following steps are taken:

1. Calculate the first derivative: f'$x$ = 9x² - 12x
2. Set the first derivative to zero and solve: 9x² - 12x = 0
3. Factor and solve for x: x$9x - 12$ = 0, giving x = 0 or x = 4/3
4. Calculate the second derivative: f''$x$ = 18x - 12
5. Evaluate the second derivative at the critical points to determine the nature of the extrema

Example: At x = 0, f''(0) = -12 < 0, indicating a local maximum. At x = 4/3, f''$4/3$ = 12 > 0, indicating a local minimum.

For Wendepunkte, the process involves:

1. Set the second derivative to zero: 18x - 12 = 0
2. Solve for x: x = 2/3
3. Verify that the third derivative is non-zero at this point

Highlight: The y-coordinates of the critical points are calculated by plugging the x-values back into the original function.

The page concludes with the identification of the following points:

• Hochpunkt (HP): (0, 2)
• Tiefpunkt (TP): $4/3, -7/9$
• Wendepunkt (WP): $2/3, 14/27$

Vocabulary:

• Hochpunkt (HP): Local maximum point
• Tiefpunkt (TP): Local minimum point
• Wendepunkt (WP): Inflection point

This comprehensive approach demonstrates the power of calculus in analyzing function behavior and identifying critical points, which is essential in various fields of mathematics and its applications.