Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 3x + 5y

Verified step by step guidance

1. Identify the vertices of the feasible region. These are the points where the boundary lines of the region intersect.
2. Substitute each vertex into the objective function \(3x + 5y\) to calculate the value of the objective function at each vertex.
3. Compare the calculated values of the objective function at each vertex to determine which one is the maximum and which one is the minimum.
4. The vertex with the highest value of the objective function is where the maximum occurs, and the vertex with the lowest value is where the minimum occurs.
5. Ensure that all calculations are consistent with the constraints that define the feasible region.

Key Concepts

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, typically in the form of a linear equation. In this case, the objective function is 3x + 5y, which we aim to maximize or minimize. Understanding how to interpret and manipulate this function is crucial for determining optimal values of the variables involved.

Feasible Region

The feasible region is the set of all possible points that satisfy the constraints of an optimization problem. It is typically represented graphically as a polygon on a coordinate plane. Identifying this region is essential because the maximum and minimum values of the objective function can only occur at the vertices of this region.

Linear Programming

Linear programming is a method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It involves maximizing or minimizing a linear objective function subject to linear constraints. Familiarity with linear programming techniques, such as the graphical method, is necessary for solving the problem effectively.