where a, k and e are positive constants. determine p' (.13) and p\"(sc) Does p(;i:) have a maximum? Sketch a graph of p[;r:). 5. Suppose the tax T a person pays 011 income to is given by T(w) = a(bw + e)\" + kw, where a, b: c, p and k are positive constants, and p > 1. (a) Find the average tax rate function The} = E. (b) Find the value of income 10 that minimizes the average tax rate. 1. A rm's production function is QUE) = 12152 $153, where L denotes the number of workers, with L E [0, 200]. (a) What size of the work force maximizes output QUL)? (b) What size of the workforce (say L') maximizes output per worker %? [Hintz Dene a function y(L) = Q(L)J'L. Maximize y(L). } (c) Show that Q'(L') : gLLl Why? 2. By producing and selling Q units of some commodity, a rm earns a total revenue R052) 2 0.0016Q2 + 4462 and incurs a cost of C(Q) 2 0.0004(92 + so + 64000. (a) What production level maximizes prots? (b) The elasticity of C(Q) with respect to Q is estimated to be 0.12 for Q : 1000. Interpret this result in words. 3. Let f(:r) : (e23 + 4e_\"')2. (a) Find f'(a:) and f\"(:r:). (1)) Determine where f (.r) is increasing and decreasing, and show that f is convex. (c) Find possible global extreme points (maxima or minima) for f. 4. In an economic model, the proportion of families whose income is no more than :0, and who have a home computer, is given by p(:c) = a + k(l 6\")