**Solution:**

To find the extremum value of √(x²y²) subject to the constraint 13x² - 10xy + 13y² = 72, we will use the method of Lagrange multipliers.

**Step 1: Formulating the Problem**

We start by defining the function f(x, y) = √(x²y²), which represents the quantity we want to optimize. We also have the constraint function g(x, y) = 13x² - 10xy + 13y² = 72.

**Step 2: Setting up the Lagrange Multiplier Equation**

We introduce a Lagrange multiplier λ and set up the Lagrange multiplier equation:

∇f = λ∇g

where ∇f and ∇g are the gradients of f and g, respectively.

The gradient of f is given by:

∇f = (∂f/∂x, ∂f/∂y) = (2xy²/√(x²y²), 2x²y/√(x²y²))

The gradient of g is given by:

∇g = (∂g/∂x, ∂g/∂y) = (26x - 10y, 26y - 10x)

**Step 3: Solving the Lagrange Multiplier Equation**

By equating the components of the gradients and multiplying by √(x²y²), we get the following system of equations:

2xy² = λ(26x - 10y)
2x²y = λ(26y - 10x)
13x² - 10xy + 13y² = 72

Simplifying the first two equations, we have:

13xy² - 5xyy = 13λx² - 5λxy
13x²y - 5xyx = 13λy² - 5λxy

Combining like terms, we get:

13xy² - 5xyy - 13λx² + 5λxy = 0
13x²y - 5xyx - 13λy² + 5λxy = 0

Factoring out common terms, we obtain:

xy(13x - 5y - 13λx + 5λ) = 0
xy(13y - 5x - 13λy + 5λ) = 0

Since xy cannot be zero, we have two possibilities:

1. 13x - 5y - 13λx + 5λ = 0
2. 13y - 5x - 13λy + 5λ = 0

**Step 4: Solving the System of Equations**

We solve the system of equations to find the values of x, y, and λ that satisfy the conditions. Adding and subtracting the two equations, we get:

(13 - 13λ)x + (5λ - 5)y = 0
(13 - 13λ)y + (5λ - 5)x = 0

Simplifying, we have:

(13 - 13λ)(x - y) = 0
(13 - 13λ)(y - x) = 0

This

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