Question

Convince yourself that the maximum and minimum values of a linear function \(f(x, y)=a x+b y+c\) over a closed polygonal set (i.e., a polygon and its interior) will always occur at a vertex of the polygon. Then use this fact to find each of the following: (a) maximum value of \(2 x+3 y+4\) on the closed polygon with vertices \((-1,2),(0,1),(1,0),(-3,0)\), and \((0,-4)\) (b) minimum value of \(-3 x+2 y+1\) on the closed polygon with vertices \((-3,0),(0,5),(2,3),(4,0)\), and \((1,-4)\)

Short answer

(a) Max value 8 at (-1,2). (b) Min value -11 at (4,0).

Step-by-step solution

Understanding Linear Functions on Polygons

A linear function of the form \( f(x, y) = ax + by + c \) is a plane in three-dimensional space. When constrained to a polygon, it has constant rates of change and does not curve, so any maximum or minimum occurs at the vertices.

Evaluating Function at Vertices for Maximum

To find the maximum value of the function \( 2x + 3y + 4 \) over the polygon with vertices \((-1,2), (0,1), (1,0), (-3,0), (0,-4)\), evaluate the function at each vertex:- \( f(-1, 2) = 2(-1) + 3(2) + 4 = 8 \)- \( f(0, 1) = 2(0) + 3(1) + 4 = 7 \)- \( f(1, 0) = 2(1) + 3(0) + 4 = 6 \)- \( f(-3, 0) = 2(-3) + 3(0) + 4 = -2 \)- \( f(0, -4) = 2(0) + 3(-4) + 4 = -8 \)The maximum value is 8 at the vertex \((-1,2)\).

Evaluating Function at Vertices for Minimum

To find the minimum value of the function \(-3x + 2y + 1 \) over the polygon with vertices \((-3,0), (0,5), (2,3), (4,0), (1,-4)\), evaluate the function at each vertex:- \( f(-3, 0) = -3(-3) + 2(0) + 1 = 10 \)- \( f(0, 5) = -3(0) + 2(5) + 1 = 11 \)- \( f(2, 3) = -3(2) + 2(3) + 1 = -1 \)- \( f(4, 0) = -3(4) + 2(0) + 1 = -11 \)- \( f(1, -4) = -3(1) + 2(-4) + 1 = -10 \)The minimum value is -11 at the vertex \((4, 0)\).

Key concepts

Polygonal Sets

A polygonal set is essentially a 2D figure comprised of straight lines connecting vertices, forming a shape with an interior. This could be anything from a simple triangle to a complex pentagon. In mathematical terms, it’s a region in a plane enclosed by a finite number of straight line segments. These segments form a closed loop or circuit.
Understanding polygonal sets is crucial when analyzing linear functions over specific domains. When a linear function is constrained within a polygonal set, it takes on interesting properties at each of its vertices. Since a linear equation represents a flat plane or a surface in three-dimensional space, the function does not curve between these points.

• Closed Polygonal Sets: Include all the interior points of the polygon.
• Vertices: Key points where the segments of a polygon meet. These are important as any maximum or minimum value of a linear function over a polygonal set occurs at these vertices.

The interaction of linear functions and polygonal sets is a foundation for understanding optimization problems in geometry or applied fields.

Vertex Evaluation

Evaluating vertices when working with linear functions is a systematic process. Since a linear function changes at consistent rates along the straight edges of a polygonal set, it naturally finds its extreme values—maximum and minimum—at the vertices.
This simply means we need to calculate the function's value at each vertex. Let's break down how:

• Select each vertex: Determine the set of vertices for the given polygon.
• Plug in coordinates: For each vertex, substitute the x and y coordinates into the linear function.
• Evaluate function: Calculate the value to see how high or low it gets at these points.

By performing these evaluations, one can definitively identify the points where the function achieves its greatest and least values within the given region. This strategy utilizes the predictable nature of linear functions to simplify solving maximum and minimum value problems.

Maximum and Minimum Values

When dealing with linear functions, the process of determining the maximum and minimum values within a polygonal set becomes straightforward due to the nature of the line. These extremities are reached at the various vertices of the polygon.
Here’s a brief step-by-step:

• Evaluate all vertices: As explained earlier, calculate the linear function at each vertex.
• List values: Write down the function's value at each vertex clearly.
• Examine extremes: Identify which of these values is the largest (maximum) and smallest (minimum).

This method ensures a methodical approach. It rules out the need to assess within the interior of the polygon, as for linear functions over polygonal domains, only vertices matter for extreme values. Knowing this helps solve applied problems where these values define critical constraints or main objectives.

Three-Dimensional Space

A linear function of two variables, like the ones in this context, can be visualized as a plane in three-dimensional space. These planes are flat and extend indefinitely in three directions. The equation of the linear function shapes these planes.
Understanding it in three-dimensional terms gives insight into how these functions behave geometrically, which is particularly useful when constrained in a polygonal set. Each point \(x, y\) on a polygon translates into a unique point \(x, y, f(x, y)\) on the plane.

• Visualizing the Plane: Comprehend that each point on this geometric figure represents possible solutions of the function.
• Effect of Coefficients: The values of \(a\), \(b\) and \(c\) in \(f(x, y) = ax + by + c\) decide the steepness and tilt of the plane.
• Geometry of Linear Functions: The combination of these coefficients and variables influence how the plane interacts with the polygonal set.

Recognizing these principles aids in connecting algebraic calculations with spatial reasoning, enhancing the understanding and graphical interpretation of linear functions further.