Determine whether the statement is true: The maximum of

f(x, y) = 5x - Зу

subject to the constraint

x² + y² 2 = 136

is 68.

To determine if the statement is true, we need to find the maximum value of the function \( f(x, y) = 5x - 3y \) subject to the constraint \( x^2 + y^2 = 136 \).

### Step 1: Identify the Lagrange Multipliers Method
We can use the method of Lagrange multipliers, where we introduce a new variable \(\lambda\) and set up the system:

\[ \nabla f = \lambda \nabla g \]

where \( g(x, y) = x^2 + y^2 - 136 = 0 \) is the constraint function.

### Step 2: Calculate the Gradients
The gradients are:
- \( \nabla f = (5, -3) \)
- \( \nabla g = (2x, 2y) \)

So, the system of equations is:
\[ 5 = 2x\lambda \]
\[ -3 = 2y\lambda \]
\[ x^2 + y^2 = 136 \]

### Step 3: Solve the Equations
1. From \( 5 = 2x\lambda \) and \( -3 = 2y\lambda \), we can solve for \(\lambda\):
   \[ \lambda = \frac{5}{2x} \]
   \[ \lambda = \frac{-3}{2y} \]

   Equating the two expressions for \(\lambda\):
   \[ \frac{5}{2x} = \frac{-3}{2y} \]
   \[ 5y = -3x \]
   \[ x = -\frac{5}{3}y \]

2. Substitute \( x = -\frac{5}{3}y \) into the constraint:
   \[ \left(-\frac{5}{3}y\right)^2 + y^2 = 136 \]
   \[ \frac{25}{9}y^2 + y^2 = 136 \]
   \[ \frac{34}{9}y^2 = 136 \]
   \[ y^2 = \frac{136 \times 9}{34} \]
   \[ y^2 = 36 \]
   \[ y = \pm 6 \]

3. Using \( y = 6 \) or \( y = -6 \) to find \( x \):
   - For \( y = 6 \):
     \[ x = -\frac{5}{3} \times 6 = -10 \]
   - For \( y = -6 \):
     \[ x = -\frac{5}{3} \times (-6) = 10 \]

### Step 4: Evaluate \( f(x, y) \) at Critical Points
- At \( (x, y) = (-10, 6) \):
  \[ f(-10, 6) = 5(-10) - 3(6) = -50 - 18 = -68 \]

- At \( (x, y) = (10, -6) \):
  \[ f(10, -6) = 5(10) - 3(-6) = 50 + 18 = 68 \]

### Conclusion
The maximum value of \( f(x, y) \) subject to the constraint is indeed 68, which occurs at \( (10, -6) \). Therefore, the statement is **true**.

---

Would you like more details on the Lagrange multipliers method or any other aspect of the solution?

Here are 5 questions you might be interested in:

1. How do Lagrange multipliers work in optimization problems?
2. What are some alternative methods for solving constrained optimization problems?
3. Can Lagrange multipliers be used for more than two variables?
4. How do you verify that a point is a maximum or minimum in constrained optimization?
5. What are common pitfalls when applying Lagrange multipliers?

**Tip:** Always check the sign and evaluate the function at all critical points to ensure you find both maximum and minimum values.

Discover how to maximize the function f(x, y) = 5x - 3y subject to the constraint x² + y² = 136 using Lagrange multipliers. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation