Optimization Problems with Constraints

This page focuses on Extremwertaufgaben mit Nebenbedingungen (optimization problems with constraints), a crucial application of calculus in real-world problem-solving.

The process for solving these problems is outlined in four steps:

1. Describe the target variable to be optimized.
2. Identify the constraints that show relationships between variables.
3. Determine the objective function that depends on only one variable.
4. Analyze the objective function, considering its domain and constraints.

Example: A problem involving maximizing the area of a rectangle with a fixed perimeter is used to illustrate this process.

The objective function is derived as A(a) = a(25-a), where 'a' is one side of the rectangle and the constraint is that the perimeter is 50 cm.

Highlight: The domain of the function is crucial and is determined based on the physical constraints of the problem.

The necessary condition for extrema is found by setting the derivative of the objective function to zero: A'(a) = 0.

Vocabulary: The hinreichende Bedingung (sufficient condition) for a maximum is that A"(a) < 0 at the critical point.

The solution process demonstrates how to incorporate the constraint into the objective function, solve for the optimal value, and verify it using the sufficient condition.

This approach to Extremwertprobleme (optimization problems) showcases the practical application of calculus in solving real-world problems, particularly those involving geometric or physical constraints.

Calculating Extreme Points

This page focuses on the methods for Extrempunkte berechnen (calculating extreme points) of functions.

The necessary condition for extreme points is that the first derivative of the function equals zero: f'(x) = 0. This is demonstrated through two examples.

Example: For f(x) = x³ - 3x², the first derivative is f'(x) = 3x² - 6x. Setting this to zero and solving yields potential extreme points.

The sufficient condition for extreme points involves examining the second derivative:

• If f"(x) > 0, it's a local minimum
• If f"(x) < 0, it's a local maximum

Highlight: The sign change test (Vorzeichenwechselkriterium) is used to determine the nature of extreme points when f"(x) = 0.

A more complex example is provided with f(x) = -1/8x⁴ + 1/3x³ + 1, demonstrating the process of finding and classifying extreme points.

Vocabulary: A saddle point (Sattelpunkt) occurs when f'(x) = 0 and f"(x) = 0, but there's no extreme value.

The page concludes with a detailed analysis of the function's behavior around its critical points, including the slope and concavity.