Page 3: Detailed Solutions

This page provides detailed solutions to the problems presented on the previous pages.

Solution to Problem 1

The solution demonstrates the step-by-step process for analyzing the function family fa(x) = a·x² + 3x² - 2.

a) Symmetry analysis shows that the function is even (symmetric about the y-axis) for all real values of a.

b) For a = -1, the inflection points are calculated: W₁ (2|8) and W₂ (-2|8)

c) The slope at point P(2|3) for a = -3 is calculated as m = 15/11

Highlight: This solution exemplifies the thorough approach required in Kurvendiskussion Aufgaben mit Lösungen Klasse 11 PDF, showing each step of the calculation process.

Solution to Problem 2

The solution to the flu outbreak problem is presented in detail:

a) The maximum point (peak of the outbreak) is calculated to occur at approximately 15.94 days, with about 129,004 students sick at that time.

b) The point of steepest decline (where the number of sick students is decreasing most rapidly) is determined to be around 25.02 days.

Example: This solution demonstrates how to apply calculus concepts to analyze real-world scenarios, a key aspect of Kurvendiskussion Aufgaben mit Lösungen klasse 10 PDF.

Page 4: Continuation of Solutions

This page continues with solutions to the remaining problems.

Solution to Problem 3

The correctness of the mathematical statements is evaluated:

a) The statement is incorrect. For a local extremum at x=b, f'(b) must equal 0, not be non-zero.

b) The statement is false. Cubic functions always have at least two extrema (one maximum and one minimum).

c) The statement is correct. Every saddle point is indeed an inflection point, as it represents a change in concavity.

Definition: An inflection point is a point on a curve at which the curvature changes sign, from concave upwards to concave downwards or vice versa.

Solution to Problem 4

The solution for the function family f(x) = x³ - 4t·x² + 4t²·x is provided:

a) For t = -1, the zeros are calculated: x₁ = 0, x₂ = 2, x₃ = -2

b) To have a zero at x₁ = 6, t must equal 3.

Vocabulary: Nullstellen berechnen (calculating zeros) is a fundamental skill in function analysis, crucial for understanding the behavior of functions.

Solution to Problem 5

The solution demonstrates how to determine the parameter t for the function f(x) = x³ + 3x² + 3tx + 1 to have exactly two points with horizontal tangents:

The value of t is calculated to be -3, and an example function satisfying this condition is provided.

Highlight: This solution showcases the application of Funktionsschar Parameter bestimmen techniques to achieve specific function characteristics.