point at (0, 0). (B) 40. (A) Show that (0, 0) is a critical point of the function 8 (x, y) = ery + x y' + 2, but that the second- derivative test for local extrema fails. (B) Use cross sections, as in Example 2, to decide whether 45. Mini g has a local maximum, a local minimum, or a saddle point at (0, 0). P(x, Sis Isiistem sesel S enluper How fell and to the Find cost Applications 41. Product mix for maximum profit. A firm produces two types of earphones per year: x thousand of type A and y thou- sand of type B. If the revenue and cost equations for the year are (in millions of dollars) R ( x, y ) = 2x + 3y C(x, y) = x2 - 2xy + 2y2 + 6x -9y + 5 determine how many of each type of earphone should be produced per year to maximize profit. What is the maximum 46. Minim profit? nates o 42. Automation-labor mix for minimum cost. The annual 47. Minim labor and automated equipment cost (in millions of dollars) paralle for a company's production of HDTVs is given by cubic i C(x, y) = 2x2 + 2xy + 3y2 - 16x - 18y + 54 materi where x is the amount spent per year on labor and y is the amount spent per year on automated equipment (both in mil- lions of dollars). Determine how much should be spent on each per year to minimize this cost. What is the minimum cost? 43. Maximizing profit. A store sells two brands of camping chairs. The store pays $60 for each brand A chair and $80 for each brand B chair. The research department has estimated the weekly demand equations for these two competitive products to be x = 260 - 3p + q Demand equation for brand A 48. Minim