Complete the following problems: 1. For the function f(x) = 4x3 - 24x2 - 60x (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. 2. Determine the open intervals on which the graph of y -3x 3 + 8x 2 + 6 x - 8 is concave downward or concave upward. Indicate any inflection points. 3. 4. Find two positive numbers such that the sum of the first and four times the second is 56 and whose product is a maximum. What is the product? Find the maximum area of a rectangle with a perimeter of 60 meters. 5. Locate the absolute extrema of the function on the closed interval [-24,24]: 24 x f ( x) 2 x 16 6. Find a function f that has derivative f ( x ) -6 x - 5 and with a graph passing through the point (-4,3). 7. Find the point of inflection of the graph of the function f(x) = cos(x/6) on the interval 0,12 . 8. Find all relative extrema of the function f(x) = 2x3 - 6x + 3. Use the Second Derivative Test to determine whether each extremum is a maximum or a minimum. 9. 3000t 2 , 0 t 3, Suppose the annual sales S of a new product is given by S 2 t2 where t is time in years. Find the exact time when the annual sales are increasing at the greatest rate. Round your answer to three decimal places. 10. Find the following limits: lim x 3x 25 x 2 - 7 -7 x - 5 x -8 x + 6 lim -4 x - 7 x 6 x 2 + 4 lim lim x -5 x - 1 9x2 - 6 x 1 lim 8 - 8 x x