Suppose you have a bounded feasible region with corner points (0, 7),
(3, 4),(9, 1), and (12, 0). Find the minimum of the objective function C = 3x + 4y
on this feasible region, as well as where it occurs.
The minimum value of C is ? and occurs at (x, y) = ?.


We are given the objective function \( C = 3x + 4y \), and we need to minimize this function over a bounded feasible region with corner points \((0, 7)\), \((3, 4)\), \((9, 1)\), and \((12, 0)\).

To find the minimum value of the objective function \( C \), we evaluate \( C \) at each corner point of the feasible region.

### 1. At \( (0, 7) \):
\[
C = 3(0) + 4(7) = 0 + 28 = 28
\]

### 2. At \( (3, 4) \):
\[
C = 3(3) + 4(4) = 9 + 16 = 25
\]

### 3. At \( (9, 1) \):
\[
C = 3(9) + 4(1) = 27 + 4 = 31
\]

### 4. At \( (12, 0) \):
\[
C = 3(12) + 4(0) = 36 + 0 = 36
\]

### Conclusion:
The minimum value of \( C \) occurs at \( (3, 4) \), where \( C = 25 \).

### Final Answer:
- The minimum value of \( C \) is \( 25 \).
- The minimum occurs at \( (x, y) = (3, 4) \).

Would you like more details on this process or a breakdown of the geometry? 

Here are 5 questions you can explore further:
1. How does linear programming relate to corner points in optimization?
2. What is the role of the feasible region in a linear programming problem?
3. How can you visualize the objective function \( C = 3x + 4y \) graphically?
4. What are the properties of convex polygons in relation to optimization?
5. How would the problem change if the objective function was \( C = 2x + y \)?

**Tip**: In linear programming, the minimum or maximum value of the objective function always occurs at a corner point of the feasible region.

Suppose you have a bounded feasible region with corner points (0, 7), (3, 4), (9, 1), and (12, 0). Find the minimum of the objective function C = 3x + 4y on this feasible region, as well as where it occurs. The minimum value of C is ? and occurs at (x, y) = ?.

Explore how to minimize the objective function C = 3x + 4y over a bounded feasible region with corner points (0, 7), (3, 4), (9, 1), and (12, 0). This step-by-step solution covers linear programming concepts and optimization techniques, making it ideal for students in Grades 10-12.

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