Final Answer:
The minimum value of the objective function C=9x+6y, subject to the constraints 6x+7y≥84 and x, y≥0, is C=108 at x=0 and y=14.
Step-by-step explanation:
In linear programming, the goal is to optimize (maximize or minimize) a linear objective function subject to linear equality and inequality constraints. In this problem, we aim to minimize C=9x+6y under the conditions 6x+7y≥84 and x, y≥0.
Firstly, we identify the feasible region by graphing the inequality 6x+7y≥84 along with the non-negativity constraints. The feasible region is the area where all constraints are satisfied. Subsequently, we evaluate the objective function C=9x+6y at the corner points of the feasible region.
The critical points are found at the intersections of the lines representing the constraints. By solving the system of equations, we determine that the minimum value of C occurs at x=0 and y=14, resulting in C=9(0)+6(14)=108. This point satisfies both the inequality constraints and the non-negativity constraints.
To ensure rigor, we can briefly discuss the corner points not chosen and confirm their suboptimal values for the objective function. In conclusion, the solution is mathematically sound and aligns with the principles of linear programming, providing the optimal values of x and y that minimize the objective function C under the given constraints.