a). The polar coordinates, x = rose and y = rsin 0 changes f (x, y) to g (r, 0) (i). Find [Verify using Mathematica] (ii). Given that of of of of -=1, evaluate ax av ax2 at the point (r, 0) = 2, 2 . [Verify using Mathematical (8 + 2 = 10 marks] b). Consider the function f (x, y) = -(x2-1) -(x y-x-1 ). (1). Find the critical points of f (x, y). [Verify using Mathematica] (ii). Using the Second Partials Test (SPT), find the relative extrema's of f (x, y). [Verify using Mathematica] (iii).Does f violate the Extreme Value Theorem? Indicate reason. [3 + 6 + 1 = 10 marks] c). As a result of the water crisis in Lasa, the Lasa Department of Water, with the assistance of their engineers and accountants decided to build a water line that has to be cost effective. The water line will be built from point A to D, as shown in Figure 3. The cost differs, given as such: 3k from A to B, 2k from B to C, and k from C to D, where the cost per kilometer is in dollars (for simplicity, let k = 1). The accountants worked together with the engineers to minimize the total cost. Use Lagrange multipliers to find x, y and z such that the total cost C will be minimized. [Verify using Mathematica]A 2km B P - - ----- 1km C D X V Z 10km Figure 3: The water line plan. [10 marks] d). Determine the dimensions of a rectangular box, open at the top, having volume V, and requiring the least amount of material for its construction. Use: (i). The constraint to eliminate a variable (Second Partials Test (SPT)). [Verify using Mathematica] (ii). Lagrange multipliers. [Verify using Mathematica] [5 + 5 = 10 marks]