Extra Credit Problems for Test 3 Problem 1: True or False (just write true or false, no justication needed): 1. If f (x) is differentiable for all values of 3:, then f(a:) must attain a maximum value on any interval [(1,1)]. 2. If f'(m) > 0 for all values of 1', then x) > 0 for some values of 1'. 3. If f(::c) is continuous on [tab], dierentiable on (a, b), and f(a) = f(b), then the Mean Value Theorem tells us that there must be c E (a, b) such that f'(c) = D. 4. If f (:r) is continuous on [(1, b], and differentiable on (a, b), then Rolle's Theorem tells us that there must be c E (a, b) such that f'(c) = 0. Problem 2: Given f(:r) = eES + coszr on the interval [3', 'r:1\"], determine the following: 1. Critical points (list any/ all as ordered pairs) Absolute Maximum Absolute Minimum Interval(s) on which the function is increasing Interval(s) on which the function is decreasing Inection points (list any/ all as ordered pairs) Interval(s) on which the function is concave up 9074991990.\" Interval(s) on which the function is concave down Problem 3: Assume 5 S f($) S 5 for :5 6 [3,8], and 6 S f'(:r) g 4 for all values of I. What is the maximum possible value of | f (3) f(8)|? Is it possible to nd the maximum possible value of |(f(3))2 (f(8))2|? If yes, nd the maximum possible value. If no, give a reason why. Problem 4: A 50ft ladder is placed against a large building. The base of the ladder is resting on an oil spill, and it slips at a rate of 3 ft /min. Find the rate of change of the height of the top of the ladder above the ground at the instant when the base of the ladder is 30 ft from the base of the building. Problem 5: The radius of a cylinder is increasing at a rate of 1 meter per hour, and the height of the cylinder is decreasing at a rate of 4 meters per hour. At a certain instant, the base radius is 5 meters and the height is 8 meters. What is the rate of change of the volume of the cylinder at that instant? Problem 6: Verify that In (1 + :5) z a at a = 0. Problem 7: Sketch the graph of the function f(:r) given \\/2+1 , 1: 221 f(x)= xi\". \"mm. \"WW- Use the guidelines for curve sketching and show your work for each step