GMC is producing cars using machines (K) and labor (L). The technology is capital intensive; the production function is given by F(K, L) = K(3/4)L(1/2). The value of GMC physical capital (machines, real estate etc.) is equal to K =$16 billon (in calculations ignore billions). We analyze rm's behavior in the short run (note K cannot be changed in the short run). Suppose the price of a car is equal to p and the wage rate is w (parameters). a) Write down the profit as a function of L.

b) On a graph with L on the horizontal axis and $ on the vertical axis, plot two components of proit function: total revenue (pF(K, L)), and labor cost (wL) (when drawing, assume p = 1 and w = 2). On the graph, mark the prot level as the dierence between the two lines (for any given L).

c) In order to find x that maximizes some function f(x), we take the rst derivative of the function with respect to x and set it equal to zero (we call it a rst order condition). Please explain intuitively why this method allows us to find the optimum.