1. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: Every local maximum is a global maximum. (b) True or False: Every global minimum is a local minimum. (c) True or False: If f has a global maximum at r = 2 on the interval (-00, 0), then the global maximum of f on the interval [0, 4] must also be at x = 2. (d) True or False: If f has a global maximum at r = 2 on the interval [0, 4], then the global maximum of f on the interval (-0o, co) must also be at x = 2. (e) True or False: If f is continuous on an interval I, then f has both a global maximum and a global minimum on I. (f) True or False: Suppose f has two local minima on the interval [0, 10], one at x = 2 with a value of 4 and one at x - 7 with a value of 1. Then the global minimum of f on [0, 10] must be at x - 7. (g) True or False: N f has no local maxima on (-0o, co), then it will have no global maximum on the interval [0, 5]. (h) True or False: If f'(3) - 0, then f has either a local minimum or a local maximum at x = 3. 2. Examples: Construct examples of the thing(s) described in the following. (a) The graph of a function with a local minimum at r = 2 but no global minimum on [0, 4). (b) The graph of a function with no local or global extrema on (-3, 3) (c) The graph of a function whose global maximum on [2, 6] does not occur at a critical point. 3. Find two real numbers x and y whose sum is 36 and whose product is as large as possible. 4. Find two real numbers x and y whose sum is 36 and whose product is as small as possible. 5. Find real numbers a and b whose sum is 100 and for which the sum of the squares of a and b is as small as possible