Question

Maximize \( Z = 5x + 3y \) subject to \( x + y \le 6 \) and \( x, y \ge 0 \). Where does the maximum value occur?

• A. \( (0,0) \)
• B. \( (0,6) \)
• C. \( (6,0) \)
• D. \( (3,3) \)

Hint:

In two-variable LPP problems, always evaluate the objective function only at the corner points of the feasible region.

Answer 1

The Correct Option is C

Concept: In Linear Programming Problems, the optimum value occurs at the corner points of the feasible region.

Step 1: Find the feasible region vertices.
The constraints are: \[ x+y\le6,\quad x\ge0,\quad y\ge0 \] Thus, the corner points are: \[ (0,0),\ (6,0),\ (0,6) \]

Step 2: Evaluate the objective function.
At \((0,0)\): \[ Z=5(0)+3(0)=0 \] At \((6,0)\): \[ Z=5(6)+3(0)=30 \] At \((0,6)\): \[ Z=5(0)+3(6)=18 \]

Step 3: Determine the maximum value.
The maximum value is: \[ 30 \] which occurs at: \[ (6,0) \]