Problem 6. Show that the function f(x) = 13 - 3x + 2 satisfies the mean the theorem. value theorem on the interval [-2, 2], then find all values of c which satisfy Problem 7. Show that the function f(r) = sin () satisfies the Rolle's theorem. theorem on the interval [7, "], then find all values of c which satisfy the Problem 8. Consider the function f(x) = 21 - 212 + 3 (a) Use a sign chart to find the intervals of increase / decrease for f(x). (b) Find the relative maximums and minimums for f (x). (c) Use a sign chart to find the intervals of concave up/ down for f(x). (d) Find any points of inflection that exist. Problem 9. For the function f(x) = xle- do parts (a) through (d) from problem 8 Problem 10. Find each of the following limits. 1. lim 2 + 12 co 1 - 2x2 1 - sin 0 2. lim r- 1 + cos 20 In r graph of f'(2) is Blow 3. lim in for . I-70+ local 4. lim I-H 5. lim (1 + 3x)2/1 6. lim x tan (1/2) 2Problem 11. Consider the function y = Vx2 + x - I 1. Find all intervals of increase and decrease. 2. Find all relative max/min 3. Find all intervals of concave up and down 4. Find all points of inflection 5. Now graph the function on the axes below labeling all the points you found. -1 -3 -2 - Problem 12. If f(-1) = 2, f'(-1) = 0 and f"(-1) = -0.1, is x = -1 a local min for f(x), a local max, or possibly neither? Draw a brief sketch to justify your conclusions. Problem 13. Sketch an example of a function that is concave up on (-co, 2), concave down on (2, co), decreasing on (-0o, 0) and (4, co), and increasing on (0, 4). Problem 14. The graph of f'(x) is below. Use this graph to determine the intervals of increase/ decrease/concave up/ concave down for f(x). At which coordinates does f(r) have a local min, local max, or inflection point? f' (I) - 2 - 1 2 3 1 -2 3