How Derivatives SECTION 4.3 8. The graph of the first derivative f' of a function f is shown. (a) On what intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or minimum? Explain. (c) On what intervals is f concave upward or concave down- ward? Explain. (d) What are the x-coordinates of the inflection points of f Why? 27. f'(0) f'(2) =f'(4) f'(x)> 0 if x< 0 or f'(x)<0 if 0 0 if 1 < x< 28. f'(x)> 0 for all x f"(x)> 0 if x< 1 o 29. f'(5) 0. f'(x) f'(x)> 0 when x > f" (x)< O when x f"(x)> 0 for 2 < x y= f(x) 30. f'(0)= f'(4) 0, f'(x)>O if 0 0 if -1 < f"(x) < 0 if x > 4 31. f'(x)> 0 if x 2, f"(x) < 0 if x > 2 lin 9. f(x) x3- 3x2-9x + 4 10. f(x) 2x3-9x2+12x - 3 lim f(x) 8, 1 11. f(x) x -2x2 +3 X 12. f(x) = x2 1 32. Suppose f(3) for all x. 2- 13. f(x)= sin x + cos x, 0 x< 2T 14. f(x) cosx 2 sin x, 0 x< 2T (a) Sketch a possi (b) How many so Why? (c) Is it possible 15. f(x)= e2x + e 16. f(x)= x2 In x 17. f(x) x - x- Inx 18. f(x) xte 33. Suppose f is a co f(0)= 4, f'(x) if 0 x< 2, f x <-1 or x> 1 (a) Can f have a graph of f. If (b) Can f have a graph of f. I (c) Sketch a pos absolute mi 19-21 Find the local maximum and minimum values off using both the First and Second Derivative Tests. Which method do you prefer? 20. f(x) 19. f(x) 1 +3x2- 2x3 21. f(x) /x 22. (a) Find the critical numbers of f(x) = x*(x - 1) (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you? 34. The graph of a f are the followin 23. Suppose f" is continuous on (-co, oo), (a) If f'(2)= 0 and f"(2)= 5, what can you say about f? (b) If f'(6) 0 and f"(6) = 0, what can you say about f? d2 dy and dx (a) dx 1 d2 y and (b) dx dx 24-31 Sketch the graph of a function that satisfies all of the given conditions. is nega (c) dx 24. (a) f'(x)<0 and f"(x)<0 for all x (b) f'(x)> 0 and f"(x)> 0 for all x yA 25. (a) f'(x)> 0 and f"(x) < 0 for all x (b) f'(x)< 0 and f"(x)> 0 for allx f'(x)> O if x < -2, 26. Vertical asymptote x 0, f'(x)-2 (x 0), f "(x)< 0 if x< 0, f"(x)>0 if x > 0 0