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).