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To solve the given optimization problem, we need to find the minimum and maximum values of the objective function \( f(x, y) = 6x + 9y \) subject to the constraints:

- \( x \geq 0 \) (non-negative values of \( x \)),
- \( y \geq 0 \) (non-negative values of \( y \)),
- \( x + y \leq 8 \) (sum of \( x \) and \( y \) cannot exceed 8).

### Steps:
1. **Identify the boundary of the feasible region**: The constraints define a triangular region in the first quadrant of the coordinate plane, with vertices determined by the following:
   - \( x = 0, y = 0 \) (origin),
   - \( x = 8, y = 0 \) (on the \( x \)-axis),
   - \( x = 0, y = 8 \) (on the \( y \)-axis),
   - Line \( x + y = 8 \) determines the third side of the triangle.

2. **Evaluate the objective function at the vertices**:
   - At \( (0, 0) \), \( f(0, 0) = 6(0) + 9(0) = 0 \),
   - At \( (8, 0) \), \( f(8, 0) = 6(8) + 9(0) = 48 \),
   - At \( (0, 8) \), \( f(0, 8) = 6(0) + 9(8) = 72 \).

3. **Evaluate along the line \( x + y = 8 \)**: 
   The points along this line are of the form \( (x, 8 - x) \). Substituting into the objective function:
   \[
   f(x, 8 - x) = 6x + 9(8 - x) = 6x + 72 - 9x = -3x + 72.
   \]
   This is a decreasing linear function, so the minimum occurs at the largest \( x \), and the maximum at the smallest \( x \). Evaluating at the endpoints:
   - At \( (8, 0) \), \( f(8, 0) = 48 \) (already computed),
   - At \( (0, 8) \), \( f(0, 8) = 72 \) (already computed).

### Conclusion:
- The **minimum value** of the objective function is 0, occurring at \( (0, 0) \).
- The **maximum value** of the objective function is 72, occurring at \( (0, 8) \).

Do you want more details or have any questions?

### 5 Follow-up Questions:
1. How would the solution change if the objective function were different, say \( f(x, y) = 7x + 5y \)?
2. Can we apply this method if the region defined by the constraints is not a triangle?
3. What happens if one of the inequalities is strict, like \( x + y < 8 \)?
4. How would the problem change if we added an additional constraint, like \( x \leq 5 \)?
5. How do we handle optimization problems when the feasible region is unbounded?

### Tip:
When solving linear programming problems geometrically, always check the vertices of the feasible region, as these give the optimal values.

Find the minimum and maximum values of the objective function f(x, y) = 6x + 9y, subject to the constraints x >= 0, y >= 0, and x + y <= 8.

Solve the optimization problem of maximizing and minimizing f(x, y) = 6x + 9y subject to x >= 0, y >= 0, and x + y <= 8. Learn how to use linear programming techniques to find solutions at the vertices.

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