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:
- Calculate the first derivative: f'(x) = 9x² - 12x
- Set the first derivative to zero and solve: 9x² - 12x = 0
- Factor and solve for x: x9x−12 = 0, giving x = 0 or x = 4/3
- Calculate the second derivative: f''(x) = 18x - 12
- 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:
- Set the second derivative to zero: 18x - 12 = 0
- Solve for x: x = 2/3
- 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.
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