3.3 Increasing and Decreasing Functions and the First Derivative Test 187 3.3 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CK Intervals on Which a Function Is Increasing or Decreasing In Exercises 11-18, find the open intervals on which the function is increasing or decreasing. 1. Increasing and Decreasing Functions Describe the Test reasing and Decreasing Functions in your CONCEPT ity, ion et. Own wo 11. g(x) x2- 2x 8 2. First Derative Test Describe the First Derivative 12. h(x) 12x - Test in yuwn words. 13. y x/16 - nd 9 14. y x + - Using a Graph in Exercises 3 and 4, use the graph off to find (a) the larEet open interval on which f is increasing and (b) the largest open interval on which f is decreasing. 15. f(x)= sin x - 1, 0< x < 2T Зx 16. f(x) = cos 0 < x < 2 2 17. y x- 2 cos x, 0 < x < 27T 4. У 3. 18. f(x) sin2 x +sin x, 0 < x < 27t 6 10 8 Applying the First Derivative Test In Exercises 19-40, (a) find the critical numbers of f, if any, (b) find the open intervals on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your 2 6 4+ 2 2- х 4 6 8 10 results. Using a Graph In Exercises 5-10, use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. 19. f(x) x2-8x 20. f(x) x2 + 6x + 10 21. f(x) =-2x2 4x + 3 22. f(x) -3x2- 4x - 2 23. f(x) =-7x3 + 21x + 3 24. f(x) x3- 6x2 +15 25. f(x) (x - 1)2(x +3) 26. f(x) = (8- x)(x + 1)2 6. f(x) x2- 6x + 8 5. y (x + 1)2 x5-5x -x6 + 6x 27. f(x) = 5 28. f(x) = у у 10 4 + X 29. f(x) x1/3 + 1 30. f(x) x2/3 - 4 -3 1 -1 -1 3 32. f(x) (x 3)1/3 34. f(x) x + 3| - 1 31. f(x) = (x + 2)2/3 2 -2 33. f(x) 5- x - 5 1 -3 x 1 х 4 5 2 35. f(x) 2x + 36. f) = 1 AT x-5 X x2- 2x + 1 38. f(x) = 37. f(x) = 8. f(x) xt- 2r 7. y= Зx x2-9 (2x + 1, x s -1 2-2, x> -1 4 -x, x 0 У у 40. f(x) 11 39. f(x) = AV -2x, Applying the First Derivative Test In Exercises 41-48, consider the function on the interval (0, 2). (a) Find the open intervals on which the function is increasing or decreasing. (b) Apply the First Derivative Test to identify all relative extrema. (c) Use a graphing utility to confirm your results. 2 + -2 -2 4 X + -2 2 1 9. flx) 10. y 2x 1 x+1) 42. f(x) = sin x cos x + 5 41. f(x) x- 2 sin x 44. f(x) =+cosx 43. f(x) = sin x + cos x 46. fx) = sinx -cos x sin x 45. f) cos (2x) 48. flx) = - 1 + cos?x 47. f(x) sinx + sin x -4-3-2-1 FORATED FOR 4 17 2