Answer
See the explanation
Work Step by Step
To solve max-min problems, follow these general steps:
1. Understand the Problem: Carefully read the problem to understand what needs to be maximized or minimized and what constraints are given.
2. Define Variables: Identify the variables involved and define them clearly. Assign symbols to represent the quantities involved.
3. Write the Objective Function: Formulate the objective function to be maximized or minimized using the defined variables.
4. Set up Constraints: Identify any constraints or limitations imposed on the variables. Write them as equations or inequalities.
5. Optimization: Use mathematical techniques such as differentiation, substitution, or graphical methods to optimize the objective function within the constraints.
6. Check for Extrema: Once you find critical points or solutions, check whether they correspond to maximum or minimum values by using the first or second derivative tests, or by examining the objective function over the feasible region.
7. Interpret the Results: Interpret the optimized values in the context of the problem to provide meaningful solutions.
Example:
Problem: A rectangular garden is to be constructed using 100 meters of fencing. Find the dimensions of the garden that maximize its area.
Solution:
1. Understand the Problem: We need to maximize the area of a rectangular garden given a fixed amount of fencing.
2. Define Variables: Let \( x \) be the length and \( y \) be the width of the rectangular garden.
3. Write the Objective Function: The objective is to maximize the area, which is given by \( A = xy \).
4. Set up Constraints: The perimeter of the garden is given by \( 2x + 2y = 100 \), or \( x + y = 50 \).
5. Optimization: Solve the constraint equation for one variable and substitute it into the objective function. We get \( y = 50 - x \). Substitute this into the objective function: \( A = x(50 - x) = 50x - x^2 \).
6. Check for Extrema: Differentiate \( A \) with respect to \( x \) to find critical points: \( \frac{dA}{dx} = 50 - 2x \). Set it equal to zero to find critical points: \( 50 - 2x = 0 \Rightarrow x = 25 \). Check the second derivative to confirm that this is a maximum.
7. Interpret the Results: The maximum area occurs when the length \( x = 25 \) meters and the width \( y = 50 - 25 = 25 \) meters. So, the dimensions of the garden that maximize its area are \( 25 \) meters by \( 25 \) meters.
