5128_Ch04_pp186-260

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Section 4.1 Extreme Values of Functions 193 EXPLORATION 1 Finding Extreme Values Let f x x x 2 1 , 2 x 2. 1. Determine graphically the extreme values of f and where they occur. Find f at these values of x. 2. Graph f and f or NDER f x, x, x in the same viewing window. Comment on the relationship between the graphs. 3. Find a formula for f x. Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.) In Exercises 1–4, find the first derivative of the function. 1 1. f x 4 x 24 x 2 2x 2. f x 9 x 2 (9 x ) 3/2 3. gx cos ln x sin ( ln x) 4. hx e 2x 2e 2x x In Exercises 5–8, match the table with a graph of f (x). 5. 6. (c) x fx (b) x fx a 0 a 0 b 0 b 0 c 5 c 5 7. 8. (d) x fx (a) a does not exist b 0 c 2 x fx a does not exist b does not exist c 1.7 a b c (c) In Exercises 9 and 10, find the limit for 2 f x . 9 x 2 9. lim f x 10. lim f x x→3 x→3 In Exercises 11 and 12, let x 3 2x, x 2 f x { x 2, x 2. a b c 11. Find (a) f 1, 1 (b) f 3, 1 (c) f 2. Undefined (d) 12. (a) Find the domain of f . x 2 (b) Write a formula for f x. f (x) 3x2 2, x 2 1, x 2 a b c (a) a b c (b) Section 4.1 Exercises In Exercises 1–4, find the extreme values and where they occur. 1. y 2. y 2 –2 0 2 x 1 –1 –1 1. Minima at (2, 0) and (2, 0), maximum at (0, 2) 2. Local minimum at (1, 0), local maximum at (1, 0) 1 x 3. y 4. 5 0 2 x y (1, 2) 2 –3 2 –1 3. Maximum at (0, 5) 4. Local maximum at (3, 0), local minimum at (2, 0), maximum at (1, 2), minimum at (0, 1) x
194 Chapter 4 Applications of Derivatives In Exercises 5–10, identify each x-value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem. See page 195. 5. y 6. y y h(x) y f(x) 0 a c 1 c 2 b 7. y 8. y f(x) x 0 a c b y y h(x) x Group Activity In Exercises 31–34, find the extreme values of the function on the interval and where they occur. 31. f x x 2 x 3, 5 x 5 32. gx x 1 x 5, 2 x 7 33. hx x 2 x 3, x 34. kx x 1 x 3, x In Exercises 35–42, identify the critical point and determine the local extreme values. 35. y x 23 x 2 36. y x 23 x 2 4 37. y x4 x 2 38. y x 2 3 x 39. 4 2x, x 1 y { x 1, x 1 0 a c b 9. y 10. y g(x) 0 a c b x x 0 a b In Exercises 11–18, use analytic methods to find the extreme values of the function on the interval and where they occur. See page 195. y c y g(x) 0 a c b x x 40. 3 x, x 0 y { 3 2x x 2 , x 0 x 41. 2 2x 4, x 1 y { x 2 6x 4, x 1 { 1 4 x2 1 2 x 1 5 , x 1 42. y 4 x 3 6x 2 8x, x 1 43. Writing to Learn The function Vx x10 2x16 2x, 0 x 5, models the volume of a box. 11. f x 1 (a) Find the extreme values of V. Max value is 144 at x 2. ln x, 0.5 x 4 x (b) Interpret any values found in (a) in terms of volume of 12. gx e x , 1 x 1 the box. The largest volume of the box is 144 cubic units and it occurs when x 2. 13. hx ln x 1, 0 x 3 44. Writing to Learn The function 14. kx e x 2 , x Px 2x 15. f x sin ( x p 4 ) 20 0 , 0 x , , 0 x 7 x 4p models the perimeter of a rectangle of dimensions x by 100x. 16. gx sec x, p 2 x 3 (a) Find any extreme values of P. Min value is 40 at x 10. 2p 17. f x x 25 , 3 x 1 (b) Give an interpretation in terms of perimeter of the rectangle 18. f x x 35 for any values found in (a). The smallest perimeter is 40 units , 2 x 3 and it occurs when x 10, which makes it a 10 by 10 square. In Exercises 19–30, find the extreme values of the function and where Standardized Test Questions they occur. Min value 1 at You should solve the following problems without using a 19. y 2x 2 8x 9 x 2 20. y x 3 2x 4 graphing calculator. 21. y x 3 x 2 8x 5 22. y x 3 3x 2 3x 2 None 45. True or False If f (c) is a local maximum of a continuous Min value 0 23. y x 2 1 function f on an open interval (a, b), then f (c) 0. Justify your 1 at x 1, 1 24. y x 2 Local max at (0, 1) 1 answer. 1 Min value 1 46. True or False If m is a local minimum and M is a local maximum of a continuous function f on (a, b), then m M. Justify 25. y 26. y Local min 1 x 2 1 at x 0 3 1 x 2 at (0, 1) your answer. 27. y 3 2x x 2 Max value 2 at x 1; 28. y 3 min value 0 at x 1, 3 47. Multiple Choice Which of the following values is the absolute maximum of the function f (x) 4x x 2 6 on the interval 2 x4 4x 3 9x 2 10 [0, 4]? E x x 1 29. y x 2 30. y 1 x 2 2x 2 (A) 0 (B) 2 (C) 4 (D) 6 (E) 10 21. Local max at (2, 17); local min at 4 3 , 4 1 27 45. False. For example, the maximum could occur at a corner, where f(c) would not exist.
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