We are tasked with finding the absolute minimum and maximum of the function $f(x, y) = 4 - 7x + 11y$ on a closed triangular region with vertices at $(0, 0)$, $(11, 0)$, and $(11, 15)$.

Steps to Solve:

1. Check the function at the vertices: Since the function is linear, its extreme values will occur either at the vertices or along the edges of the triangular region.
2. Analyze the edges of the triangle: We will examine the function on each edge of the triangle.

1. Check the vertices:

• At $(0, 0)$, $f(0, 0) = 4 - 7(0) + 11(0) = 4$.
• At $(11, 0)$, $f(11, 0) = 4 - 7(11) + 11(0) = 4 - 77 = -73$.
• At $(11, 15)$, $f(11, 15) = 4 - 7(11) + 11(15) = 4 - 77 + 165 = 92$.

2. Analyze the edges:

• Edge 1: Between $(0, 0)$ and $(11, 0)$:

  • Along this edge, $y = 0$, so $f(x, 0) = 4 - 7x$.
  • The function decreases as $x$ increases, and the minimum occurs at $(11, 0)$ where $f(11, 0) = -73$, and the maximum occurs at $(0, 0)$ where $f(0, 0) = 4$.

• Edge 2: Between $(11, 0)$ and $(11, 15)$:

  • Along this edge, $x = 11$, so $f(11, y) = 4 - 7(11) + 11y = -73 + 11y$.
  • The function increases as $y$ increases, so the minimum occurs at $(11, 0)$ where $f(11, 0) = -73$, and the maximum occurs at $(11, 15)$ where $f(11, 15) = 92$.

• Edge 3: Between $(0, 0)$ and $(11, 15)$:

  • The equation of this edge is $y = \frac{15}{11}x$, so substitute this into $f(x, y)$: $f\left(x, \frac{15}{11}x\right) = 4 - 7x + 11\left(\frac{15}{11}x\right) = 4 - 7x + 15x = 4 + 8x.$
  • This function increases as $x$ increases, so the minimum occurs at $(0, 0)$ where $f(0, 0) = 4$, and the maximum occurs at $(11, 15)$ where $f(11, 15) = 92$.

Conclusion:

• Minimum value: $-73$
  Occurs at: $(11, 0)$
• Maximum value: $92$
  Occurs at: $(11, 15)$

Would you like more details on any part of the process?

Here are 5 related questions to think about:

1. How would the solution change if the triangular region had different vertices?
2. What happens if we use a different linear function for $f(x, y)$?
3. Can a linear function have multiple minima or maxima in a triangular region?
4. How would this method apply to non-linear functions on a triangular region?
5. How would you verify the result using Lagrange multipliers?

Tip: For linear functions like this, the extreme values often occur at the boundary or vertices.