Question

The maximum value of the objective function $z=2x+3y$, when the corner points of the feasible region are (0, 0), (5, 0), (4, 1) and (0, 2), is:

• A. 0
• B. 6
• C. 10
• D. 11
• E. 16

Hint:

In Linear Programming, the optimal solution always occurs at one of the corner points.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a problem in linear programming. The Fundamental Theorem of Linear Programming states that the optimal (maximum or minimum) value of a linear objective function over a convex polygonal feasible region will always occur at one of the corner points (vertices) of that region.
Step 2: Key Formula or Approach:
We will evaluate the objective function \( z = 2x+3y \) at each of the given corner points of the feasible region. The largest value obtained will be the maximum value.
Step 3: Detailed Explanation:
The objective function is \( z = 2x+3y \).
The corner points are (0,0), (5,0), (4,1), and (0,2).
Let's evaluate z at each point:
- At point (0, 0):
\[ z = 2(0) + 3(0) = 0 \] - At point (5, 0):
\[ z = 2(5) + 3(0) = 10 + 0 = 10 \] - At point (4, 1):
\[ z = 2(4) + 3(1) = 8 + 3 = 11 \] - At point (0, 2):
\[ z = 2(0) + 3(2) = 0 + 6 = 6 \] Now, we compare the values of z obtained: \{0, 10, 11, 6\}.
The largest value among these is 11.
Step 4: Final Answer:
The maximum value of the objective function is 11.