Question

If \( x \), \( y \), and \( z \) are all positive and \( x + y + z = 9 \), the maximum value of \( xyz \) is ............

Hint:

When maximizing the product of variables under a linear constraint, the maximum occurs when the variables are equal.

Answer 1

Correct Answer: 27

Step 1: Use the method of Lagrange multipliers.
We are given the constraint \( x + y + z = 9 \), and we need to maximize the product \( xyz \). Using symmetry and the fact that \( x = y = z \) for the maximum (since the product of variables is maximized when they are equal under a linear constraint), we can set: \[ x = y = z \] Substituting this into the constraint \( x + y + z = 9 \), we get: \[ 3x = 9 \quad \Rightarrow \quad x = 3 \] Step 2: Calculate the maximum value.
Substituting \( x = 3 \), we get the maximum value of \( xyz \) as: \[ xyz = 3 \times 3 \times 3 = 27 \] Step 3: Conclusion.
Thus, the maximum value of \( xyz \) is \( \boxed{27} \).