Problem. 1 : Use the method of Lagrange multipliers to minimize the function subject to the given constraint. f(x, y) = x' + 4 y' subject to xy = 1 at (x, y) = ( ? minimum of larger x-value minimum of ? at (x, y) = ( ? smaller r-value Problem. 2 : Use the method of Lagrange multipliers to maximize the function subject to the given constraint. f(x, y) = 4x + 4y - 2x' - 6 y' subject to 8r + 6 y = 7 at (x, y) = ( ? maximum of Problem. 3 : Use the method of Lagrange multipliers to maximize the function subject to the given constraints. f(x, y) = ry' subject to 4x3 + y' = 4 and y 2 0 maximum of at (x, y) = ( ? Problem. 4 : Use the method of Lagrange multipliers to maximize the function subject to the given constraints. f(x, y) = ry' subject to x? + y? = 36 and y 2 0 maximum of at (x, y) = ( ? minimum of ? at (x, y) = ( Problem. 5 : Use the method of Lagrange multipliers to maximize the function subject to the given constraint. f(x, y, z) = ryz subject to 10x + 10 y + = = 84 and > > 0,y 2 0, = 2 0 maximum of at (x, y, z) = ( ?Problem. 6 : Gator FanStu'manulacures at sells 2 tailgate canopies and g,- coolers per week. The total weekly prot (in dollars} realized from thae sales is given by the function P[x, y) = 41.152\" 0.20 y2 0.15 my + 702 + Tl] y3BDU. The companyr decides to restrict the rate of manufacture of these items to a total of 180 per week (grand total of both items). Determine how many canopies and coolers the company should manufacture each week to maximize its profit.(H.ound down to the nearest whole unit; note that the total may then be less than 180.) - canopies coolers Problem. 7 : Gator Bar and Grill has decided to enclose 1330 ft? area outside their building to provide outdoor seating. The enclosed area will he arectangle, one side of which is provided by the external wall of the building. Two sides will he made of pine board and the remaining side will he made of steel fencing material. lfthe pine hoard fencing costs $ 3 per linear foot and the steel fencing costs $ 2 per linear foot, determine the dimensions of the enclosure that will cost the least to build. (Round your answers to two decimal places.) ? each pine side I ft steel side ft Problem. 3 : Gator Golf Carts has determined that it talres m units of labor and 3; units of capital to produce z, y} units of its product, where aw) = 5'] 1'\" y\"- lfa unit of labor costs ii 6|), :1 unit of capital costs$ 180, ands 150000 is budgeted for production, determine how many units should be expended on labor and how many units should be expended on capital in order to maximise production. (Round your answers to two decimal places.) labor units capital units Problem. 9 :Let z = :1:2 szi 63,2 and suppose that (2,51) changes from (2, 1) to (1335,1115). (Round your answers to four decimal places.) (3) Compute Az. (b) Compute dz