To solve this function or optimize it, we will need to address it step-by-step. Let's break down the given function:

\[
L = 0.8 \log X + 0.2 \log Y + \lambda (5000 - 4X - 2Y)
\]

where \( X \), \( Y \), and \( \lambda \) (likely a Lagrange multiplier) are variables.

### Step 1: Lagrange Multiplier
The given expression looks like it involves the method of **Lagrange multipliers** for optimization under a constraint. The constraint seems to be \( 5000 - 4X - 2Y = 0 \), based on the \(\lambda\) term.

If you want to **maximize** or **minimize** \(L\), you'll need to:

### Step 2: Partial Derivatives
To apply the method of Lagrange multipliers, you should take the partial derivatives of \(L\) with respect to \(X\), \(Y\), and \(\lambda\), and set them equal to zero to solve for the critical points.

1. **Partial derivative with respect to \(X\):**

\[
\frac{\partial L}{\partial X} = \frac{0.8}{X} - 4\lambda = 0
\]

2. **Partial derivative with respect to \(Y\):**

\[
\frac{\partial L}{\partial Y} = \frac{0.2}{Y} - 2\lambda = 0
\]

3. **Partial derivative with respect to \(\lambda\):**

\[
\frac{\partial L}{\partial \lambda} = 5000 - 4X - 2Y = 0
\]

### Step 3: Solving the System of Equations
You now have a system of three equations:

1. \(\frac{0.8}{X} = 4\lambda\)
2. \(\frac{0.2}{Y} = 2\lambda\)
3. \(5000 - 4X - 2Y = 0\)

We can solve these equations to find \(X\), \(Y\), and \(\lambda\).

#### From Equation (1):
\[
\lambda = \frac{0.8}{4X} = \frac{0.2}{Y}
\]
This gives the relationship \(Y = 4X\).

#### Substituting \(Y = 4X\) into Equation (3):
\[
5000 - 4X - 2(4X) = 0
\]
\[
5000 - 12X = 0
\]
\[
X = \frac{5000}{12} = 416.67
\]

#### Substituting \(X = 416.67\) into \(Y = 4X\):
\[
Y = 4(416.67) = 1666.67
\]

### Step 4: Conclusion
The optimal values of \(X\) and \(Y\) are approximately:
\[
X \approx 416.67, \quad Y \approx 1666.67
\]
You can find \(\lambda\) from these values by substituting into one of the earlier equations.

---

Would you like further clarification on any step? Here are 5 related questions to extend this topic:

1. How do we apply the method of Lagrange multipliers to other optimization problems?
2. What is the significance of logarithmic functions in optimization problems?
3. How do you interpret \(\lambda\) in this type of optimization problem?
4. Could we solve this numerically if exact solutions weren't possible?
5. How do logarithmic derivatives help in economics or growth models?

**Tip**: Always check the constraints carefully in optimization problems to ensure you're not missing any additional limitations.

L = 0.8logX + 0.2logY + λ(5000 - 4X - 2Y)

Solve the optimization problem L = 0.8logX + 0.2logY + λ(5000 - 4X - 2Y) using Lagrange multipliers and logarithmic functions. This detailed step-by-step solution involves partial derivatives and solving a system of equations. Suitable for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation