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:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and D only
By the corner point theorem, if the feasible region R of an LPP is bounded, the objective function z = ax + by has both a maximum and a minimum value on R, and each occurs at a corner point of R. Only for an unbounded region may a maximum or minimum fail to exist.
So on the bounded region R, z has both a maximum and a minimum, and both occur at corner points of R.
Checking the statements
(A) The maximum or minimum value of objective function may not exist. — Incorrect. This can happen only when R is unbounded.
(B) The objective function \( z \) has both a maximum and a minimum value on R. — Correct. R is bounded, so both values exist.
(C) Maximum and minimum values lie in the unbounded region — Incorrect. R is given to be bounded, and the optimal values occur at its corner points.
(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. — Correct. This is the corner point theorem for a bounded region.
CBSE Applied Mathematics (Class XII):
Solving a Linear Programming Problem
