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To solve this problem, we need to find the maximum and minimum values of the function \( f(x, y) = x^2 - y^2 \) subject to the constraint \( g(x, y) = x^2 + y^2 - 64 = 0 \) using the method of Lagrange multipliers.

### Step-by-Step Solution

1. **Set up the Lagrange function**:
   \[
   \mathcal{L}(x, y, \lambda) = f(x, y) - \lambda g(x, y)
   \]
   Substituting in \( f(x, y) = x^2 - y^2 \) and \( g(x, y) = x^2 + y^2 - 64 \), we get:
   \[
   \mathcal{L}(x, y, \lambda) = x^2 - y^2 - \lambda (x^2 + y^2 - 64)
   \]

2. **Compute the partial derivatives and set them to zero**:
   - **Partial derivative with respect to \( x \)**:
     \[
     \frac{\partial \mathcal{L}}{\partial x} = 2x - 2\lambda x = 0 \Rightarrow x(1 - \lambda) = 0
     \]
     So, \( x = 0 \) or \( \lambda = 1 \).

   - **Partial derivative with respect to \( y \)**:
     \[
     \frac{\partial \mathcal{L}}{\partial y} = -2y - 2\lambda y = 0 \Rightarrow y(-1 - \lambda) = 0
     \]
     So, \( y = 0 \) or \( \lambda = -1 \).

   - **Partial derivative with respect to \( \lambda \)** (from the constraint):
     \[
     \frac{\partial \mathcal{L}}{\partial \lambda} = -(x^2 + y^2 - 64) = 0 \Rightarrow x^2 + y^2 = 64
     \]

3. **Analyze cases**:
   We have two cases to analyze based on values of \( x \), \( y \), and \( \lambda \):

   - **Case 1**: \( \lambda = 1 \)
     - If \( \lambda = 1 \), then \( x \) could be any value that satisfies the constraint.
     - For \( x^2 + y^2 = 64 \), and substituting into \( f(x, y) = x^2

This extreme value problem has a solution with both a maximum and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. \( f(x, y) = x^2 - y^2 \), \( x^2 + y^2 = 64 \)

This page provides a solution to finding the maximum and minimum values of the function \( f(x, y) = x^2 - y^2 \) under the constraint \( x^2 + y^2 = 64 \) using Lagrange multipliers. Step-by-step calculations involve setting up the Lagrange function, taking partial derivatives, and analyzing critical points. Suitable for college-level calculus students.

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