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This problem involves a constrained optimization problem where we need to find the "shadow price." The shadow price, also known as the Lagrange multiplier, measures the rate of change of the objective function with respect to a change in the constraint.

We are given:

- Objective function: \( y = 3x_1 x_2 + x_2^2 \)
- Constraint: \( 2x_1 + x_2 = 100 \)

### Steps to solve:

1. **Formulate the Lagrangian**: The Lagrangian function \( \mathcal{L} \) combines the objective function and the constraint, using a Lagrange multiplier \( \lambda \).

   \[
   \mathcal{L}(x_1, x_2, \lambda) = 3x_1 x_2 + x_2^2 + \lambda(100 - (2x_1 + x_2))
   \]

2. **Take the partial derivatives**: To find the optimal solution, take the partial derivatives of the Lagrangian with respect to \( x_1 \), \( x_2 \), and \( \lambda \), and set them equal to zero.

   - \( \frac{\partial \mathcal{L}}{\partial x_1} = 3x_2 - 2\lambda = 0 \)
   - \( \frac{\partial \mathcal{L}}{\partial x_2} = 3x_1 + 2x_2 - \lambda = 0 \)
   - \( \frac{\partial \mathcal{L}}{\partial \lambda} = 100 - (2x_1 + x_2) = 0 \)

3. **Solve the system of equations**: We now solve these three equations to find \( x_1 \), \( x_2 \), and \( \lambda \).

Let's solve the system:

From the first equation, solve for \( \lambda \):

   \[
   \lambda = \frac{3x_2}{2}
   \]

Substitute this into the second equation:

   \[
   3x_1 + 2x_2 - \frac{3x_2}{2} = 0
   \]

Multiply through by 2 to eliminate the fraction:

   \[
   6x_1 + 4x_2 - 3x_2 = 0
   \]

Simplify:

   \[
   6x_1 + x_2 = 0
   \]

Thus,

   \[
   x_1 = -\frac{x_2}{6}
   \]

Now substitute this into the constraint equation \( 2x_1 + x_2 = 100 \):

   \[
   2\left(-\frac{x_2}{6}\right) + x_2 = 100
   \]

Simplify:

   \[
   -\frac{x_2}{3} + x_2 = 100
   \]

Multiply through by 3 to eliminate the fraction:

   \[
   -x_2 + 3x_2 = 300
   \]

Thus,

   \[
   2x_2 = 300 \quad \Rightarrow \quad x_2 = 150
   \]

Now, substitute \( x_2 = 150 \) back into the equation for \( x_1 \):

   \[
   x_1 = -\frac{150}{6} = -25
   \]

4. **Find the shadow price \( \lambda \)**: Use \( x_2 = 150 \) in the expression for \( \lambda \):

   \[
   \lambda = \frac{3(150)}{2} = 225
   \]

Thus, the shadow price (or Lagrange multiplier) is \( \lambda = 225 \).

---

Would you like further details on this method or have any specific questions?

Here are 5 related questions for deeper exploration:
1. How does the value of \( \lambda \) change if the constraint changes slightly?
2. Can we interpret the shadow price in an economic context?
3. What is the significance of the negative value for \( x_1 \) in this problem?
4. How do we solve the same problem using numerical methods or software tools?
5. What if the constraint were nonlinear? How would this affect the solution method?

**Tip**: The shadow price gives insight into how sensitive the objective function is to changes in the constraint. Understanding this can help in resource allocation problems.

What is the shadow price in the following constrained optimization problem? Maximize y = 3x1x2 + x2^2, subject to 2x1 + x2 = 100.

Learn how to calculate the shadow price (Lagrange multiplier) for the constrained optimization problem involving y = 3x1x2 + x2^2, subject to the constraint 2x1 + x2 = 100. Detailed step-by-step solution using Lagrange multipliers.

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