Chapter 12 — Linear Programming

NCERT Class 12 Maths Chapter 12, Linear Programming, teaches how to find the optimal (maximum or minimum) value of a linear objective function subject to linear constraints, solved using the graphical Corner Point Method.

Chapter 12 of NCERT Class 12 Mathematics Part II introduces Linear Programming as a special class of optimisation problems. Students learn to formulate real-life problems — such as maximising profit or minimising cost — as linear functions subject to constraints. The graphical method is used exclusively: the feasible region (a convex polygon) is identified, and the objective function is evaluated at each corner point to find the optimal value. Key theorems establish that optimal values always occur at corner points of the feasible region.

1. 01A Linear Programming Problem seeks the optimal value (maximum or minimum) of a linear objective function Z = ax + by subject to linear constraints and non-negative restrictions.
2. 02The feasible region is the common region satisfying all constraints; it is always a convex region and every point in it is a feasible solution.
3. 03By Theorem 1, the optimal value of the objective function must occur at a corner point (vertex) of the feasible region.
4. 04By Theorem 2, if the feasible region is bounded, the objective function has both a maximum and a minimum, each occurring at a corner point.
5. 05The Corner Point Method involves finding all vertices of the feasible region, evaluating Z at each, and selecting the largest or smallest value; if two corner points yield the same optimal value, every point on the segment joining them is also optimal.
6. 06If the feasible region is unbounded, a maximum or minimum may not exist; this must be verified by checking whether the open half-plane beyond the candidate value shares points with the feasible region.