Question

Let R be the feasible region for a linear programming problem, and Let \(z=ax+by\) be the objective function. If the feasible region R is bounded then which of the following are true ?
A. The maximum or minimum value of objective function may not exist.
B. The objective function \(z\) has both a maximum and a minimum value on R.
C. Maximum and minimum values lie in the unbounded region
D. The objective function \(z=ax+by\) has both a maximum value and a minimum value and each of these values occurs at a corner point of R.
Choose the correct answer from the options given below:

• A. A and B only
• B. A and D only
• C. B and D only
• D. C and D only

Hint:

Bounded feasible region: max and min both exist, at corner points.

Answer 1

The Correct Option is C

Step 1: Think about why bounded matters.
A linear function is continuous, and a bounded feasible region defined by linear inequalities is a closed and bounded polygon. A continuous function on a closed bounded set always reaches a largest and a smallest value.

Step 2: Where the extremes occur.
Because \(z=ax+by\) is linear, it cannot have a peak in the middle of the polygon. Moving in a direction that increases \(z\) can go on until an edge, then a vertex, is reached. So extremes sit at corner points.

Step 3: Sort the statements.
B and D say exactly this, so they are true. A denies existence, which is only a worry for an unbounded region, so it is false here. C places the extremes in an unbounded region, which contradicts the given bounded region, so it is false.

Step 4: Pick the option.
Only the pair B and D is correct, giving option 3.

Final Answer:
The correct pair is B and D. \[ \boxed{\text{B and D only}} \]