Question

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.

Accepted Answer

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. 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. 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. 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. 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.

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.

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Key Concepts

Graphing Constraints

Objective Function

Piecewise Functions