The Cobb–Douglas production function for a product is

N(x, y) = 10x^0.8y^0.2

where x is the number of units of labor and y is the number of units of capital required to produce N units of the product.

(A) Find the marginal productivity of labor and the marginal productivity of capital at x = 40 and y = 50. For the greatest increase in productivity, should management encourage increased use of labor or increased use of capital?

(B) If each unit of labor costs $100, each unit of capital costs $50, and $10,000 is budgeted for production of this product, use the method of Lagrange multipliers to determine the allocations of labor and capital that will maximize the number of units produced and find the maximum production. Find the marginal productivity of money and approximate the increase in production that would result from an increase of $2,000 in the amount budgeted for production.

(C) If 50 ≤ x ≤ 100 and 20 ≤ y ≤ 40, find the average number of units produced. Set up a double integral, and evaluate it.