https://file.aiquickdraw.com/mathbot/file/1727278636416tuf9n516.jpg

Let's solve the problems shown in the image step by step.

### Problem 1:
We are asked to maximize the utility function \( U(x, y) = xy \) subject to the constraint \( x + y = 10 \) using the Lagrange multiplier method.

The Lagrange function is:
\[
\mathcal{L}(x, y, \lambda) = xy + \lambda (10 - (x + y))
\]
To maximize this, we compute the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), and \( \lambda \):

1. \( \frac{\partial \mathcal{L}}{\partial x} = y - \lambda = 0 \) \(\Rightarrow \lambda = y\)
2. \( \frac{\partial \mathcal{L}}{\partial y} = x - \lambda = 0 \) \(\Rightarrow \lambda = x\)
3. \( \frac{\partial \mathcal{L}}{\partial \lambda} = 10 - x - y = 0 \)

From \( \lambda = x = y \), and the constraint \( x + y = 10 \), we have:
\[
x + x = 10 \Rightarrow 2x = 10 \Rightarrow x = 5
\]
Thus, \( x = y = 5 \) and \( \lambda = 5 \).

So the correct choice is: 
\[
\lambda = 5, x = 5, y = 5
\]
The answer is the third option.

---

### Problem 2:
The function \( f(x, y) = ax^2 + by^2 \) is said to be homogeneous of degree 2. This means that for a scaling factor \( t \), the following must hold:
\[
f(tx, ty) = t^2 f(x, y)
\]
Hence, the correct answer is the second option:
\[
f(tx, ty) = t^2 f(x, y)
\]

---

### Problem 3:
We are given a Cobb-Douglas production function \( Q = 4K^{\frac{1}{2}} L^{\frac{1}{2}} \), where the inputs \( K \) and \( L \) vary with time \( t \) and interest rate \( r \), given by:
\[
K(t, r) = \frac{10t^2}{r}, \quad L(t, r) = 6t^2 + 250r
\]
We are asked to find the rate of change of \( Q \) with respect to \( t \) when \( t = 10 \) and \( r = 0.1 \).

First, let's substitute the expressions for \( K(t, r) \) and \( L(t, r) \) into \( Q \):
\[
Q(t, r) = 4 \left( \frac{10t^2}{r} \right)^{\frac{1}{2}} \left( 6t^2 + 250r \right)^{\frac{1}{2}}
\]

We can now compute the partial derivative of \( Q \) with respect to \( t \) and evaluate it at \( t = 10 \) and \( r = 0.1 \).

Let me perform the necessary calculations.The rate of change of the Cobb-Douglas production function \( Q \) with respect to \( t \) when \( t = 10 \) and \( r = 0.1 \) is \( 1960 \).

Thus, the correct answer for Problem 3 is \( 1960 \), which is not listed among the given options. This suggests that there might be an error in the provided options.

---

Do you have any questions, or would you like further details?

### 5 Related Questions:
1. How does the Lagrange multiplier method work in optimization problems?
2. Can you explain the concept of homogeneous functions and their properties?
3. How are Cobb-Douglas production functions used in economics?
4. What is the importance of partial derivatives in optimization problems?
5. How does the interest rate affect the Cobb-Douglas production function in this example?

### Tip:
When dealing with homogeneous functions, always check if scaling both variables by a factor changes the function by the factor raised to a specific power (degree).

Solve three math problems using specific mathematical methods, including maximizing a utility function, determining the property of a homogeneous function, and calculating the rate of change in a Cobb-Douglas production function.

Explore detailed solutions to complex math problems including maximizing a utility function using Lagrangian multipliers, verifying the degree of a homogeneous function, and computing the rate change of a Cobb-Douglas production function at specific values. These solutions employ calculus and economic models suitable for university-level 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