3. Find the minimum value of 𝑥
2 + 𝑦
2 + 𝑧
2
given that 𝑥𝑦𝑧 = 8.
 


To find the minimum value of \( x^2 + y^2 + z^2 \) given that \( xyz = 8 \), we can apply some techniques from optimization, such as the **AM-GM inequality** or **Lagrange multipliers**. Here, we'll proceed using **AM-GM inequality** for simplicity.

### Step 1: Use Symmetry and AM-GM Inequality

We want to minimize \( x^2 + y^2 + z^2 \) under the constraint \( xyz = 8 \). By symmetry, let's assume that \( x = y = z \). This assumption simplifies the problem and is often a good starting point when variables appear symmetrically.

### Step 2: Simplify the Condition \( xyz = 8 \)

If \( x = y = z \), then the condition \( xyz = 8 \) becomes:
\[
x^3 = 8
\]
Solving for \( x \), we get:
\[
x = 2
\]

Since \( x = y = z \), we have:
\[
y = 2, \quad z = 2
\]

### Step 3: Calculate \( x^2 + y^2 + z^2 \)

Now, substitute \( x = 2 \), \( y = 2 \), and \( z = 2 \) into the expression \( x^2 + y^2 + z^2 \):
\[
x^2 + y^2 + z^2 = 2^2 + 2^2 + 2^2 = 4 + 4 + 4 = 12
\]

### Conclusion:

The minimum value of \( x^2 + y^2 + z^2 \) is **12**, and this occurs when \( x = y = z = 2 \).

Would you like more details on why symmetry works here or any other explanation?

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Here are 5 related questions for further practice:
1. What is the minimum value of \( x^2 + y^2 \) given that \( xy = 4 \)?
2. Solve for the minimum of \( x^2 + y^2 + z^2 \) subject to \( xyz = -8 \).
3. How can the method of Lagrange multipliers be applied to this problem?
4. Find the maximum value of \( x + y + z \) given that \( xyz = 8 \).
5. Can we find the same minimum value using derivatives instead of AM-GM?

**Tip:** The AM-GM inequality is useful for finding minimum or maximum values when you have symmetric expressions involving multiplication and addition.

Find the minimum value of x^2 + y^2 + z^2 given that xyz = 8.

Learn how to find the minimum value of x^2 + y^2 + z^2 under the constraint xyz = 8 using algebraic optimization techniques, including symmetry and the AM-GM inequality. This problem demonstrates key concepts for students in Grades 10-12.

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