Sketch the graph of the function f defined by f(x) + x2. We first find the critical numbers and possible points of inflection of f. f'(x) * x3 - 3x2 + 2x = 0 x = 0, x =1 or x =2 x(x2 - 3x + 2) = 0 f"(x) =3x2 - 6x + 2=0 x(x - 1)(x - 2) = 0 x =1# V3 3 The critical numbers are 0, 1 and 2, and the possible points of inflec- V3 V3 LION OCCUR at 2 any 21. An open box is to be made from a 20-inch by 32-inch piece of cardboard by cutting out r-inch by r-inch squares from the four corners and bending up the sides. The largest possible volume of the box is obtained by maximizing V(x) =_ for r in the interval (2-2 points) 2. A fertilizer producer finds that it can sell its product at a price of p = 300-0.1r dollars per unit when it produces a units of fertilizer. The total production cost (in dollars) for I units is C(x) = 15000 + 125x + 0.02512. If the production capacity of the firm is at most 1000 units of fertilizer in a specified time, how many units must be manufactured and sold in that time to maximize the profit? (Hint: profit-price-cost) (6 points) 3. Find S'(x) if f(r) = 2r + sin I (4 pts) 4. A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15 cin/min. At what rate must air be removed when the radius is 9 cm? (6 points)