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.