In an LPP if the objective function z = ax + by has the same maximum Value on 2 corner points of the same feasible region, then every point on the line segments joining these 2 points give the same maximum value
Turn this into graph based Ques 


Here’s how we can turn this Linear Programming Problem (LPP) into a graph-based question:

---

**Graph-Based Question:**

Consider the following linear programming problem:

Maximize the objective function \( Z = ax + by \), subject to the constraints:

1. \( x + y \leq 6 \)
2. \( 2x + y \geq 4 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Problem:
- Plot the feasible region determined by the constraints.
- Identify the corner points of the feasible region.
- If the objective function \( Z = 3x + 5y \) has the same maximum value at the corner points \( A \) and \( B \) of the feasible region, prove graphically that every point on the line segment joining \( A \) and \( B \) gives the same maximum value of \( Z \).
  
---

### Steps to Solve:

1. **Graph the constraints** on the coordinate plane and shade the feasible region.
2. **Find the corner points** of the feasible region.
3. **Check the values of the objective function \( Z = 3x + 5y \)** at the corner points.
4. Identify the points \( A \) and \( B \) where the maximum value of \( Z \) occurs.
5. Draw the **line segment joining \( A \) and \( B \)**, and show that for any point on this line, the objective function yields the same value.

---

Would you like to go through the solution steps, see how to graph this, or explore the theory behind the LPP?

### Relative Questions:
1. What does the objective function \( Z = ax + by \) represent geometrically?
2. How do we find the corner points in a linear programming problem?
3. Why does the objective function \( Z \) remain constant along the line segment between two points of the feasible region?
4. What does the feasible region represent in a graph-based LPP?
5. How do constraints affect the shape of the feasible region?

**Tip**: The maximum or minimum of a linear objective function in an LPP always occurs at a corner point or along the line joining two adjacent corner points if they share the same value.

In an LPP if the objective function z = ax + by has the same maximum value on 2 corner points of the same feasible region, then every point on the line segments joining these 2 points give the same maximum value. Turn this into a graph-based question.

Explore how the objective function Z = ax + by achieves the same maximum value at two corner points of a feasible region in a Linear Programming Problem. Understand the graphical solution method, corner points, and line segments between these points, suitable for students in Grades 11-12 or undergraduate levels.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation