Given, x and y are positive real numbers such that x + y = 4. The minimum value of the expression (x + 1/x)² + (y + 1/y)² is n. Find 4n. give step wise solution

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!

Given, x and y are positive real numbers such that x + y = 4. The minimum value of the expression (x + 1/x)² + (y + 1/y)² is n. Find 4n. Give stepwise solution.

Find the minimum value of the expression (x + 1/x)² + (y + 1/y)² under the condition that x + y = 4. This stepwise solution explores symmetry, algebraic manipulation, and optimization techniques, ideal for students in Grades 10-12.

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