During the 1920s, Charles Cobb and Paul Douglas modeled total production output P (of a firm, industry, or entire economy) as a function of labor hours involved x and capital invested y (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by P(x, y) = kx^α y^1-α, where k and a are constants representative of a particular firm or economy.

a. Show that a doubling of both labor and capital results in a doubling of production P.

b. Suppose a particular firm has the production function for k = 120 and α = 3/4. Assume that each unit of labor costs $250 and each unit of capital costs $400, and that the total expenses for all costs cannot exceed $100,000. Find the maximum production level for the firm.