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 onemaximumandoneminimum.
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 fx = 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 calculatingzeros 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 fx = 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.