The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below. The maximum value of Z - minimum value of Z is

Q: The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below. The maximum value of Z - minimum value of Z is

The correct option is **(D) 43**. **Reasoning:** Evaluating $Z = 5x + 7y$ at each corner point of the feasible region, the maximum occurs at $(3, 4)$ giving Z = 43, and the minimum occurs at the origin $(0, 0)$ giving Z = 0, so $\text{Max } Z - \text{Min } Z = 43 - 0 = 43$.

Linear Programming

Linear Programming - Feasible Region

Options

(A)8

(B)29

(D)43

Official Answer

The correct option is (D) 43.

Reasoning: Evaluating $Z = 5x + 7y$ at each corner point of the feasible region, the maximum occurs at $(3, 4)$ giving Z = 43, and the minimum occurs at the origin $(0, 0)$ giving Z = 0, so $\text{Max } Z - \text{Min } Z = 43 - 0 = 43$ .

linear programmingfeasible regionobjective functioncorner point theoremoptimal valueZ = 5x + 7ymaximum and minimum value

Marking Scheme

11 mark: correct option (D) 43 — no partial marking for MCQ.

Hint

Evaluate Z at every corner (vertex) of the shaded feasible region and subtract the smallest value from the largest.

Quick Oral Answer

By the Corner Point Theorem, the maximum and minimum of a linear objective function on a bounded feasible region occur at its vertices, so I evaluate Z at each corner and subtract the smallest from the largest.

Analysis & Explanation

This is a corner-point-theorem application on a bounded feasible region.

Why (D) is correct: By the Corner Point Theorem, the optimal (maximum/minimum) value of a linear objective function over a bounded feasible region occurs at one of its corner points. For the region shown, evaluating $Z = 5x + 7y$ at each vertex — $(0,0), (7,0), (3,4), (0,2)$ — gives Z = 0, 35, 43, and 14 respectively. The maximum is 43 (at (3,4)) and the minimum is 0 (at the origin), so $\text{Max } Z - \text{Min } Z = 43$ .

Why the distractors are wrong:

(A) 8 and (B) 29 do not correspond to any valid pair of corner-point Z-values for this region; they likely result from picking the wrong pair of vertices or an arithmetic slip.
(C) 35 is the value of Z at (7,0) alone, not the required difference — a common error is stopping after computing Z at just one boundary vertex instead of comparing all corners.

Common Mistakes

1Evaluating Z at only one or two corner points instead of all vertices of the feasible region.
2Confusing the maximum value alone (43) with the required difference ( $\text{Max } Z - \text{Min } Z$ ).
3Misreading the coordinates of a vertex from the graph, leading to an incorrect Z value.

Interesting Facts

The Corner Point Theorem (also called the Fundamental Theorem of Linear Programming) guarantees that for a bounded feasible region, the optimal value of a linear objective function always occurs at a vertex, never strictly inside the region — a fact that follows from the convexity of the feasible region.

Linear programming was developed by George Dantzig in 1947 as the Simplex Method to solve resource-allocation problems for the US Air Force; CBSE's graphical corner-point method is a simplified two-variable version of this same idea.

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Frequently Asked Questions

What is the Corner Point Theorem used to solve this MCQ?

It states that for a bounded feasible region, the maximum and minimum values of a linear objective function occur at one of the corner points (vertices) of the region, so you only need to evaluate Z at each vertex rather than every point in the region.

Why is the minimum value taken as 0 in this feasible region?

Because the origin $(0,0)$ is a vertex of this particular bounded feasible region, and $Z = 5x + 7y$ evaluates to 0 there, which is the smallest of all the corner values.