Problem 54 - Algebra 2

Question

Find the maximum and minimum values of the function \(f(x, y)=2 x+3 y\) for the polygonal region with vertices at \((2,4),(-1,3),(-3,-3),\) and $(2,-5) .

Step-by-Step Solution

Answer

The maximum value is 16, and the minimum value is -15.

Step 1: Identify the Vertices of the Polygon

The polygonal region is defined by the vertices \((2,4), (-1,3), (-3,-3), (2,-5)\). We will use these points to evaluate the function \(f(x, y) = 2x + 3y\).

Step 2: Evaluate the Function at Each Vertex

Calculate \(f(x, y) = 2x + 3y\) for each vertex:- At \((2,4)\), \(f(2, 4) = 2(2) + 3(4) = 4 + 12 = 16\).- At \((-1,3)\), \(f(-1, 3) = 2(-1) + 3(3) = -2 + 9 = 7\).- At \((-3,-3)\), \(f(-3, -3) = 2(-3) + 3(-3) = -6 - 9 = -15\).- At \((2,-5)\), \(f(2, -5) = 2(2) + 3(-5) = 4 - 15 = -11\).

Step 3: Determine Maximum and Minimum Values

From the evaluations, the function values are:- 16 at \((2,4)\)- 7 at \((-1,3)\)- -15 at \((-3,-3)\)- -11 at \((2,-5)\)The maximum value is 16, and the minimum value is -15.

Key Concepts

Vertex Evaluation

When solving linear programming problems, especially those involving a polygonal region, evaluating the vertices is key. Each vertex in a polygon represents a point of intersection of the boundary lines. In this exercise, we have vertices at

• \( (2,4) \)
• \( (-1,3) \)
• \( (-3,-3) \)
• \( (2,-5) \)

Evaluating the objective function at these points helps in finding which combinations of variables provide the best results—in terms of maximum and minimum values. Each vertex needs to be plugged into the function to see which one yields the highest or lowest value. This step is crucial as linear functions achieve their extrema at these corner points, or vertices.

Objective Function

The objective function is a crucial component of any optimization problem. It’s the mathematical expression that you want to maximize or minimize. In this problem, our objective function is \[ f(x, y) = 2x + 3y \] This linear function assigns a value based on the inputs \( x \) and \( y \). By substituting the coordinates of each vertex of the polygonal region into the function, we compute the function's value at these points. The computed values then help determine which arrangements of \( x \) and \( y \) provide the desired optimal outcome, either maximal or minimal, for the problem scenario presented.

Polygonal Region

In linear programming, a polygonal region refers to a geometric area bounded by linear inequalities. The vertices of this region are the points where these boundary lines meet. For our exercise, this region is defined by vertices

• \( (2,4) \)
• \( (-1,3) \)
• \( (-3,-3) \)
• \( (2,-5) \)

This polygonal region represents all possible solutions to the constraints which form the linear inequalities. Evaluating the vertex points within this region using the objective function provides insights into the maximum and minimum values possible within the constraints defined by the region's shape.

Maximum and Minimum Values

Discovering the maximum and minimum values of a function within a given region is the ultimate aim of linear programming. From the values calculated at each vertex of the polygonal region, we observe that:

• The maximum value is \( 16 \), occurring at the vertex \( (2,4) \).
• The minimum value is \( -15 \), which is found at \( (-3,-3) \).

These extrema are vital as they inform decision-making and optimization in real-world applications ranging from business logistics to resource management. The presence of maximum and minimum values at the vertices ensures that solutions are feasible within the given constraints.