In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

Verified step by step guidance

1. Identify the constraints provided in the problem. Constraints are inequalities that define the feasible region. For example, they might look like \(x + y \leq 10\), \(x \geq 0\), and \(y \geq 0\). Write down all the constraints clearly.
2. Graph the constraints on a coordinate plane. For each inequality, first graph the corresponding equation as a straight line (e.g., \(x + y = 10\)). Then shade the region that satisfies the inequality (e.g., for \(x + y \leq 10\), shade below the line). The feasible region is the intersection of all shaded areas.
3. Determine the vertices (corner points) of the feasible region. These are the points where the boundary lines of the constraints intersect. Solve the system of equations for each pair of boundary lines to find these points.
4. Substitute the coordinates of each vertex into the given objective function. The objective function is typically a linear function like \(z = 3x + 2y\). Evaluate the function at each vertex to determine the corresponding values.
5. Identify the maximum value of the objective function from the values calculated in the previous step. The vertex that gives the highest value is the solution to the problem, and its coordinates represent the optimal solution.

Key Concepts

Graphing Constraints

Graphing constraints involves plotting inequalities on a coordinate plane to visualize the feasible region where all constraints are satisfied. Each inequality represents a boundary, and the area where these boundaries overlap indicates the possible solutions. Understanding how to interpret and graph these inequalities is crucial for identifying the feasible region.

Objective Function

An objective function is a mathematical expression that defines the quantity to be maximized or minimized within the constraints of a problem. In optimization problems, this function is evaluated at various points within the feasible region to determine the best possible outcome. Recognizing how to formulate and analyze the objective function is essential for solving optimization problems.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value, often used to model situations where a rule changes at certain points. Understanding how to interpret and graph piecewise functions is important, as they can affect the shape of the feasible region and the evaluation of the objective function. This concept is particularly relevant when determining maximum or minimum values in optimization problems.