To solve the problem of finding the minimum value of the expression $h_1 \cdot w_1 + h_1 \cdot w_2 + h_2 \cdot w_3$, given the constraints:

$h_1 \cdot w_3 = 2$ $h_2 \cdot w_1 = 4$ $h_2 \cdot w_2 = 4$

You should focus on the following mathematical topics:

1. Optimization Techniques:

• Linear Programming: While the problem might not be linear, understanding how to set up optimization problems is crucial.
• Nonlinear Optimization: Explore how to minimize or maximize expressions with constraints, which might involve more advanced techniques like Lagrange multipliers.

2. Algebra:

• Substitution Method: This will help you substitute the values from the constraints into the expression you want to minimize.
• Quadratic Equations: Although not directly involved, understanding how to manipulate expressions will be helpful.

3. Inequalities:

• AM-GM Inequality: A specific inequality useful for determining bounds on sums and products, which can help in minimizing or maximizing expressions.

4. Calculus:

• Partial Derivatives: These are used to find the minimum or maximum of a multivariable function, especially if you eventually use calculus-based optimization.

5. Convex Analysis:

• Convex Functions: Understanding properties of convex functions can be crucial if the expression you're minimizing has a specific structure that makes it convex.

Step-by-Step Approach:

1. Substitute the given constraints into the expression $h_1 \cdot w_1 + h_1 \cdot w_2 + h_2 \cdot w_3$.

2. Simplify the expression to see if it can be expressed in terms of a single variable or a simpler function.

3. Analyze the resulting function to find its minimum, considering that all variables are non-negative.

4. Check for Boundary Values: Often, the minimum can occur at the boundaries of the allowed values for $h_1, h_2, w_1, w_2, w_3$.

If you want, I can help you with the detailed solution. Would you like to proceed with that?

Here are 5 relative questions that might expand your understanding:

1. How do you derive the constraints from an optimization problem?
2. What role do substitution and simplification play in optimization problems?
3. How do Lagrange multipliers work in constrained optimization?
4. What are the common methods to solve nonlinear equations?
5. Can the AM-GM inequality be applied directly in this context?

Tip: When dealing with optimization problems, always try to visualize the problem if possible. Sometimes graphing the equations or inequalities can provide insight into where the minimum or maximum might occur.