Question

The solution set for minimizing the function \(\mathrm{z}=x+y\) with constraints \(x+y \geqslant 2, x+2 y \leqslant 8, y \leqslant 3, x, y \geqslant 0\) contains

• A. \(x=0,y=3\)
• B. \(x=8,y=0\)
• C. infinitely many points
• D. \(x=2,y=3\)

Hint:

Find the corner points of the feasible region and evaluate z at each. If two adjacent corners give the same minimum, every point of the edge between them is optimal.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The minimum of a linear objective occurs at a corner point of the feasible region.

Step 2: Corner Points
The region bounded by \(x+y\ge2\), \(x+2y\le8\), \(y\le3\), \(x,y\ge0\) has corners \((2,0),\ (8,0),\ (2,3),\ (0,3),\ (0,2)\).

Step 3: Evaluate z = x + y
\(z\) is \(2,\ 8,\ 5,\ 3,\ 2\) at these corners respectively.

Step 4: Interpret
The minimum value 2 occurs at both \((2,0)\) and \((0,2)\). Hence every point on the segment joining them (the line \(x+y=2\)) is optimal.

Final Answer:
The minimum is attained at infinitely many points. \[ \boxed{\text{(C) }\text{infinitely many points}} \]