1. In each case, nd the relative maxima or minima of f (3,9!) subject to the given constraint. (a) Relative minimum of f(:r, y) = 4:32 +3392 my 5, Constraint: 3:: + y = 68. (b) Relative maximum of at, 9) = mzy + 89:, Constraint: :5 + y = 5. 2. Find three positive numbers say, and 2 such that zyz = 32 and my + 2072 + 2yz is maximized. 3. A manufacturing rm estimates that its total production of automobile batteries in thousands of units is EMU ay) =60wa , where :1? is the number of units of labor and y is the number of units of capital utilized. Labor costs are $120 per unit, and capital costs are $210 per unit. How many units each of labor and capital will maximize production, if the rm can spend $31500 for these costs? 4. Evaluate each double integral. [If the function seems difcult to integrate, try chang- ing the order of integration] //() 2 THE UNIVERSITY OF BRITISH COLUMBIA OKANAGAN MATH 142 - Calculus II 1 2 (b) f f .0281\" dy dz: 0 0 (c) f/m/z2+3ydzdy; Risdened byOSzglandBSyg R ((1) [/2336323' datdy; R is dened by 0 g :1: g 1 and 0 S y g 1 R 5. Find the volume under the surface 2351; Z = $23; + 1 and above the rectangle R dened by 0 S a: S 1 and 0 S y S 3. 6. The prot (in dollars) from selling :0 units of one product and y units of a second product is Pm, y) = (m 100)2 (y 50)2 + 2000. The weekly sales for the rst product vary from 100 units to 150 units, and the weekly sales for the second product vary from 40 units to 80 units. Find the average weekly prot for these two products