This question deals with second derivatives, their interpretation and application.

(a) (Conceptual portion - 10 pts) Suppose a function f(x) is defined and continuous on the real line, and is known to have the following properties: When f (5) = 2, f (5) = 0 and f (5) > 0. In addition, f (0) = 0, f (0) = 0, f (5) = 4, f (5) = 0 and f (5) < 0 where 0 is an inflection point. Draw the graph of f using the information above, and explain how you got it - in particular explain the meaning of the second derivative of a function to be positive, negative, and equal to 0. Assume there are no other critical points/inflection points of f besides those given above.

(b) (Computational Portion - 10 pts) The annual cost (in thousands of dollars) of manufacturing x thousand sets of wireless earbuds is given by C(x) = .001x3 + 3x + 100, and no more than 60000 earbuds can be produced in a year. i. Find the average cost function. ii. How many earbuds should be made in order to minimize the average cost per set? Use the second derivative test to confirm your answer is a minimum. iii. What is the minimum average cost?