Linear Programming Problem

Step 1: Profit per quintal: Rice = ₹40, Wheat = ₹25
Step 2: Objective function: Z=40x+25y

Step 1: Test each point. Substitute in constraints.
Step 2: Violation. Point (2000,0) violates x₁+x₂\le1500.

Step 1: Use the objective function formula.
The objective function represents the total interest earned from both investments. The function is given by: where is the amount invested in saving certificates and in national saving bonds.
Thus, the correct answer is (1).

Step 1: There are constraints from column sums.
Step 2: There are constraints from row sums.
Step 3: Total constraints .

On the boundary :

Maximum occurs at .

Step 1: Changing constraints changes the feasible region, hence re-evaluation is needed.

Step 1: Two optimal points on a line segment imply all convex combinations are optimal.

There are m column constraints and n row constraints. Hence, total number of constraints =m+n.

Step 1: The feasible region is bounded by: x+2y=2, x+2y=8, x=0, y=0 Step 2: Corner points of feasible region: (2,0),(8,0),(0,1),(0,4) Step 3: Evaluate z=3x+2y: z(2,0)=6, z(8,0)=24, z(0,1)=2, z(0,4)=8 Step 4: Maximum value is 24 at (8,0).

Step 1: Understanding the Concept
According to the Fundamental Theorem of Linear Programming, if an optimal solution (maximum or minimum) exists for an LPP, it must occur at one of the vertices (corner points) of the feasible region.

Step 2: Detailed Explanation
1. The Feasible Region is the set of all points that satisfy all the given constraints. 2. The Objective Function represents a family of parallel lines. 3. As we move these lines across the feasible region, the first or last point they touch before leaving the region will always be a corner point (or an entire edge if two corners give the same value). 4. Therefore, to find the minimum or maximum, we only need to test the coordinates of the corner points in the function .

Step 3: Final Answer
The minimum value occurs at the corner points of the feasible region.