Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function P(L) = 200L + 10L² - L³ gives the output of a system as a function of the number of laborers L. The average product A(L) is the average output per laborer when L laborers are working; that is A(L) = P(L)/L. The marginal product M(L) is the approximate change in output when one additional laborer is added to L laborers; that is, M(L) = dP/dL.

a. For the given production function, compute and graph P, A, and M.

b. Suppose the peak of the average product curve occurs at L = L₀, so that A'(L₀) = 0. Show that for a general production function, M(L₀) = A(L₀).