Find the absolute minimum and absolute maximum off(x,y)=4−7x+11yon the closed triangular region with vertices (0,0),(11,0) and (11,15).

List the minimum/maximum values as well as the point(s) at which they occur. If a min or max occurs at multiple points separate the points with commas.

Minimum value: 
Occurs at 

Maximum value: 
Occurs at 

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.

Find the absolute minimum and absolute maximum of f(x,y)=4−7x+11y on the closed triangular region with vertices (0,0), (11,0), and (11,15).

Discover how to find the absolute minimum and maximum of the function f(x,y)=4−7x+11y on a triangular region with vertices (0,0), (11,0), and (11,15). This problem uses key optimization techniques and multivariable calculus, making it ideal for students in Grades 11-12 or early college calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation