We are given that $x$ and $y$ are positive real numbers such that $x + y = 4$, and we need to find the minimum value of the expression:

$\left( x + \frac{1}{x} \right)^2 + \left( y + \frac{1}{y} \right)^2$

Let's denote this expression as $f(x, y)$:

$f(x, y) = \left( x + \frac{1}{x} \right)^2 + \left( y + \frac{1}{y} \right)^2$

Step 1: Expanding the squares

We first expand each square in the expression:

$\left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2}$ $\left( y + \frac{1}{y} \right)^2 = y^2 + 2 + \frac{1}{y^2}$

Thus, the expression becomes:

$f(x, y) = x^2 + 2 + \frac{1}{x^2} + y^2 + 2 + \frac{1}{y^2} = x^2 + y^2 + \frac{1}{x^2} + \frac{1}{y^2} + 4$

Step 2: Using the condition $x + y = 4$

We need to minimize the expression $f(x, y) = x^2 + y^2 + \frac{1}{x^2} + \frac{1}{y^2} + 4$, subject to the condition $x + y = 4$.

Step 3: Symmetry and intuition

Since the expression $\left( x + \frac{1}{x} \right)^2$ and $\left( y + \frac{1}{y} \right)^2$ are symmetric, the minimum value will likely occur when $x = y$. Let’s check if this assumption leads to a solution.

If $x = y$, then $x + y = 4$ implies:

$x = y = 2$

Step 4: Calculating $f(x, y)$ for $x = y = 2$

Substitute $x = y = 2$ into the expression:

$\left( x + \frac{1}{x} \right)^2 = \left( 2 + \frac{1}{2} \right)^2 = \left( \frac{5}{2} \right)^2 = \frac{25}{4}$ $\left( y + \frac{1}{y} \right)^2 = \left( 2 + \frac{1}{2} \right)^2 = \left( \frac{5}{2} \right)^2 = \frac{25}{4}$

Thus, the total value of $f(x, y)$ when $x = y = 2$ is:

$f(2, 2) = \frac{25}{4} + \frac{25}{4} = \frac{50}{4} = 12.5$

Step 5: Finding $4n$

The minimum value of the expression is $n = 12.5$. Therefore:

$4n = 4 \times 12.5 = 50$

Final Answer:

The value of $4n$ is $\boxed{50}$.

Would you like more details or have any questions? Here are 5 related questions:

1. What would happen if $x \neq y$ in this problem?
2. Can we use Lagrange multipliers to solve this problem?
3. How do we handle inequalities like $x + y = 4$ in optimization problems generally?
4. Can this problem be solved using calculus (e.g., partial derivatives)?
5. What other methods can be used to confirm the solution is minimal?

Tip: For symmetrical problems, assuming equality of variables often simplifies the process and leads to the correct answer!