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: Recall the theorem.
In linear programming, if the feasible region \(R\) is bounded, then the objective function \(Z=ax+by\) has both a maximum and a minimum value on \(R\). Each of these occurs at a corner point (vertex) of \(R\).

Step 2: Check statement A.
A says the maximum or minimum may not exist. For a bounded region this is false, because both always exist. So A is FALSE.

Step 3: Check statement B.
B says \(z\) has both a maximum and a minimum on \(R\). This is exactly the theorem. So B is TRUE.

Step 4: Check statement C.
C says the extreme values lie in the unbounded region. But \(R\) is given as bounded, and the extreme values lie at corner points of \(R\). So C is FALSE.

Step 5: Check statement D.
D says both a maximum and a minimum exist and each occurs at a corner point of \(R\). This matches the corner point theorem. So D is TRUE.

Step 6: Match with the options.
The true statements are B and D only, which is option 3.

Final Answer:
Statements B and D are true, so the answer is option 3. \[ \boxed{\text{B and D only}} \]