Solve the following Exercises (Value 10 Points) 1. Given the following data, determine the Values of K and L that optimize the Production Function: Qx = 2K + 4L + KL Y 40 = 2K + 4L

To optimize the production function Q_x = 2K + 4L + KL subject to the constraint 40 = 2K + 4L , we can use the method of Lagrange multipliers.

Define the Lagrangian function \mathcal{L}(K, L, \lambda) = 2K + 4L + KL - \lambda(2K + 4L - 40) , where \lambda is the Lagrange multiplier.

Now, find the first-order conditions by taking the partial derivatives with respect to each variable and setting them equal to zero:
\frac{\partial \mathcal{L}}{\partial K} = 2 + L - 2\lambda = 0
\frac{\partial \mathcal{L}}{\partial L} = 4 + K - 4\lambda = 0
\frac{\partial \mathcal{L}}{\partial \lambda} = 2K + 4L - 40 = 0

Solving these three equations simultaneously, we get:
L = 1
K = 10
\lambda = -\frac{1}{2}

The values of K and L that optimize the production function are K = 10 and L = 5 .

\boxed{K = 10, L = 5}