The problem you've shared involves optimizing the objective function $f(x, y) = 5x + 9y$ under the constraints:

1. $x \geq 0$
2. $y \geq 0$
3. $x + y \leq 8$

To solve this, we'll follow these steps:

1. Identifying the Feasible Region

The constraints define a triangular feasible region in the first quadrant:

• $x \geq 0$ ensures we are on or to the right of the $y$-axis.
• $y \geq 0$ ensures we are on or above the $x$-axis.
• $x + y \leq 8$ gives us a line with intercepts at $(8, 0)$ and $(0, 8)$.

This forms a triangular region with vertices at:

• $(0, 0)$
• $(8, 0)$
• $(0, 8)$

2. Evaluating the Objective Function at the Vertices

Next, we evaluate the objective function $f(x, y) = 5x + 9y$ at these vertices:

• At $(0, 0)$: $f(0, 0) = 5(0) + 9(0) = 0$
• At $(8, 0)$: $f(8, 0) = 5(8) + 9(0) = 40$
• At $(0, 8)$: $f(0, 8) = 5(0) + 9(8) = 72$

3. Checking for Other Points Along the Boundary

The line $x + y = 8$ can be parameterized as $x = 8 - y$. Substituting into the objective function: $f(x, y) = 5(8 - y) + 9y = 40 - 5y + 9y = 40 + 4y$ Thus, the value of $f(x, y)$ increases as $y$ increases. So, the maximum occurs at $y = 8$ and the minimum at $y = 0$.

4. Conclusion

• Minimum value: The minimum value of the objective function is $0$, occurring at $(0, 0)$.
• Maximum value: The maximum value of the objective function is $72$, occurring at $(0, 8)$.

Would you like further details or have any questions?

Here are 5 related questions to explore this topic further:

1. How would the solution change if the objective function was $f(x, y) = 7x + 4y$?
2. What happens if we add the constraint $x \leq 6$ to the problem?
3. Can you solve this problem graphically by plotting the objective function and constraints?
4. How do you determine if a point is inside the feasible region geometrically?
5. What are the applications of linear programming in real-world scenarios?

Tip: In optimization problems, the minimum or maximum often occurs at the boundary or corner points of the feasible region, so always check the vertices!