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The given problem is to find the maximum and minimum values of the function \( f(x, y) = xy \) subject to the constraint \( 5x + 2y = 60 \), using Lagrange multipliers.

### Step-by-step solution:

1. **Objective function**: 
   \[
   f(x, y) = xy
   \]
   
2. **Constraint**: 
   \[
   g(x, y) = 5x + 2y - 60 = 0
   \]

3. **Lagrange multiplier method**: Introduce a Lagrange multiplier \( \lambda \) and set up the system:
   \[
   \nabla f(x, y) = \lambda \nabla g(x, y)
   \]
   
   The gradients are:
   \[
   \nabla f(x, y) = (y, x)
   \]
   \[
   \nabla g(x, y) = (5, 2)
   \]
   
   This gives the system:
   \[
   y = 5\lambda \quad \text{(1)}
   \]
   \[
   x = 2\lambda \quad \text{(2)}
   \]
   
4. **Substitute into the constraint**: 
   Substitute equations (1) and (2) into the constraint \( 5x + 2y = 60 \):
   \[
   5(2\lambda) + 2(5\lambda) = 60
   \]
   Simplifying:
   \[
   10\lambda + 10\lambda = 60
   \]
   \[
   20\lambda = 60
   \]
   \[
   \lambda = 3
   \]
   
5. **Find \( x \) and \( y \)**: 
   Using \( \lambda = 3 \), we substitute back into equations (1) and (2):
   \[
   y = 5(3) = 15
   \]
   \[
   x = 2(3) = 6
   \]
   
6. **Calculate the optimal value**: 
   The optimal value of \( f(x, y) = xy \) is:
   \[
   f(6, 15) = 6 \times 15 = 90
   \]

Thus, the optimal value of \( f(x, y) \) is **90**.

Would you like further details on any step?

### Related Questions:
1. How do you interpret the Lagrange multiplier in this context?
2. What happens if the constraint changes, e.g., \( 5x + 2y = 80 \)?
3. Could there be local maxima or minima different from global ones in this problem?
4. How do we handle Lagrange multipliers if there are multiple constraints?
5. Why is it important to check both maximum and minimum values in optimization problems?

### Tip:
Always verify that the points obtained satisfy both the original function and the constraint equation to avoid errors in optimization problems.

Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = xy subject to the constraint 5x + 2y = 60, if such values exist.

Learn how to use Lagrange multipliers to find the optimal values of f(x, y) = xy under the constraint 5x + 2y = 60. Explore step-by-step solutions with key concepts from multivariable calculus, including the application of Lagrange Multiplier Theorem.

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