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.