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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!

Find the minimum and maximum values of the objective function f(x, y) = 5x + 9y, subject to the constraints x ≥ 0, y ≥ 0, and x + y ≤ 8.

Learn how to solve the linear optimization problem f(x, y) = 5x + 9y under the constraints x ≥ 0, y ≥ 0, and x + y ≤ 8. Find the minimum and maximum values, including a step-by-step evaluation of the feasible region.

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