1. **State the problem:** Minimize the function $$w = x + 2y + z$$. 2. **Understand the problem:** This is a linear function in three variables. To find its minimum, we typically need constraints (like equations or inequalities) that define the domain where $x, y, z$ can vary. 3. **Note:** Without constraints, the function can decrease indefinitely by choosing appropriate values of $x, y, z$ (e.g., making them very negative), so no minimum exists. 4. **Conclusion:** To solve this minimization problem, please provide constraints or a domain for $x, y, z$. Since no constraints are given, the problem is incomplete and cannot be solved as is.