To solve this problem using the method of Lagrange multipliers, we need to minimize the function $f(w_1, w_2, w_3, h_1, h_2)$ subject to constraints involving $w_1, w_2, w_3, h_1, h_2$.

Given:

• $w_1 h_1 = 4$
• $w_2 h_1 = 4$
• $w_3 h_2 = 2$
• $w_1 h_2 = 4$

We want to minimize: $f(w_1, w_2, w_3, h_1, h_2) = (w_1 + w_2 + w_3)(h_1 + h_2)$

This problem involves several constraints, so we will introduce Lagrange multipliers $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ for each constraint. Our goal is to find the points where the gradients of the objective function and the constraints align.

Step 1: Define the Lagrange Function

We start by forming the Lagrange function $\mathcal{L}$:

+ \lambda_1 (w_1 h_1 - 4) + \lambda_2 (w_2 h_1 - 4) + \lambda_3 (w_3 h_2 - 2) + \lambda_4 (w_1 h_2 - 4)$$ ### Step 2: Find the Partial Derivatives We now take the partial derivatives of $$\mathcal{L}$$ with respect to each variable, set them to zero, and solve the system of equations. 1. **Partial derivatives with respect to $$w_1$$, $$w_2$$, $$w_3$$, $$h_1$$, and $$h_2$$:** $$\frac{\partial \mathcal{L}}{\partial w_1} = (h_1 + h_2) + \lambda_1 h_1 + \lambda_4 h_2 = 0$$ $$\frac{\partial \mathcal{L}}{\partial w_2} = (h_1 + h_2) + \lambda_2 h_1 = 0$$ $$\frac{\partial \mathcal{L}}{\partial w_3} = (h_1 + h_2) + \lambda_3 h_2 = 0$$ $$\frac{\partial \mathcal{L}}{\partial h_1} = (w_1 + w_2 + w_3) + \lambda_1 w_1 + \lambda_2 w_2 = 0$$ $$\frac{\partial \mathcal{L}}{\partial h_2} = (w_1 + w_2 + w_3) + \lambda_3 w_3 + \lambda_4 w_1 = 0$$ 2. **The original constraint equations:** $$w_1 h_1 = 4, \quad w_2 h_1 = 4, \quad w_3 h_2 = 2, \quad w_1 h_2 = 4$$ ### Step 3: Solve the System of Equations Let's first try to solve these equations systematically: - From the constraint $$w_1 h_1 = 4$$, we can express $$w_1$$ in terms of $$h_1$$: $$w_1 = \frac{4}{h_1}$$ - From $$w_2 h_1 = 4$$, we get: $$w_2 = \frac{4}{h_1}$$ - From $$w_3 h_2 = 2$$, we get: $$w_3 = \frac{2}{h_2}$$ - From $$w_1 h_2 = 4$$, we also know that: $$w_1 = \frac{4}{h_2}$$ ### Step 4: Solve for $$h_1$$ and $$h_2$$ From $$w_1 = \frac{4}{h_1}$$ and $$w_1 = \frac{4}{h_2}$$, we conclude that: $$h_1 = h_2$$ Let $$h_1 = h_2 = h$$. Now, substitute this into the equations: $$w_1 = \frac{4}{h}, \quad w_2 = \frac{4}{h}, \quad w_3 = \frac{2}{h}$$ ### Step 5: Minimize the Objective Function The objective function becomes: $$f(w_1, w_2, w_3, h_1, h_2) = (w_1 + w_2 + w_3)(h_1 + h_2) = \left( \frac{4}{h} + \frac{4}{h} + \frac{2}{h} \right) (h + h) = \frac{10}{h} \cdot 2h = 20$$ Thus, the minimum value of the objective function is **20**. Would you like more details on any step, or any further explanation? ### 5 Related Questions: 1. What are the necessary conditions for applying Lagrange multipliers in optimization problems? 2. How would the solution change if the constraints were different? 3. Can you generalize the result to similar problems with different constants? 4. What if there were more variables or constraints—how would we approach solving it? 5. How does this method compare to other optimization methods like the method of steepest descent? ### Tip: Always check that the solution satisfies both the objective and all the constraints when using Lagrange multipliers.