Exam

Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].1) 1)

2) 2)

3) 3)

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4) 4)

5) 5)

Determine all critical numbers for the function.6) f(x) = x2 + 18x + 81 6)

7) f(x) = x3 - 12x - 5 7)

8) f(x) = x3 - 9x2 + 4 8)

9) f(x) = 80x3 - 3x5 9)

10) f(x) = (x - 6)7 10)

11) f(x) = x3 - 3x2 + 1 11)

12) f(x) = x3 - 12x + 3 12)

13) f(x) = (x + 1)3 13)

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Using the derivative of f(x) given below, determine the critical numbers of f(x).14) f (x) = (x + 7)(x + 4) 14)

15) f (x) = (x - 1)2(x + 4) 15)

Find the absolute extreme values of each function on the interval.

16) f(x) = 23

x + 2; -3 x 3 16)

17) y = 7 - 8x2 on [-4, 5] 17)

Find the absolute extreme values of the function on the interval.18) f(x) = 2x - 1, -2 x 3 18)

19) g(x) = -x2 + 9x - 18, 6 x 3 19)

20) h(x) =12 x + 4, -2 x 3 20)

21) g(x) = -x2 + 7x - 12, 3 x 4 21)

22) g(x) = 6 - 7x2, -2 x 5 22)

23) g(x) = 6 - 5x2, -2 x 3 23)

Find the location of the indicated absolute extremum for the function.24) Minimum 24)

3

25) Maximum 25)

26) Maximum 26)

27) Minimum 27)

4

28) Minimum 28)

29) Maximum 29)

30) Minimum 30)

5

31) Maximum 31)

Find the extreme values of the function and where they occur.32) y = x3 - 3x2 + 1 32)

33) y = x3 - 12x + 2 33)

34) y = x3 - 3x2 + 6x - 8 34)

35) f(x) = x2 + 2x - 3 35)

36) f(x) = x3 - 3x2 + 6x - 8 36)

37) f(x) = (x + 1)3 37)

Find the largest open interval where the function is changing as requested.

38) Increasing f(x) = 14

x2 -12

x 38)

39) Increasing f(x) = x2 - 2x + 1 39)

40) Increasing y = (x2 - 9)2 40)

41) Decreasing f(x) = x3 - 4x 41)

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.42) f (x) = (3 - x)(5 - x) 42)

Find the local extrema of the function, if they exist.43) f(x) = x2 - 6x + 18 43)

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44) f(x) = 3x2 + 6x + 1 44)

45) s(x) = -x2 - 18x + 63 45)

46) f(x) = -7x2 - 2x - 9 46)

47) f(x) = x3 - 12x + 1 47)

48) f(x) = x4 - 2x2 - 9 48)

Determine where the given function is concave up and where it is concave down.49) f(x) = x2 - 4x + 5 49)

50) q(x) = 6x3 + 2x + 3 50)

51) f(x) = 7x - x3 51)

52) f(x) = x3 + 3x2 - x - 24 52)

53) f(x) = -x3 + 6x + 2 53)

54) f(x) = x3 - 3x2 + 2x + 15 54)

55) f(x) = 2x3 + 12x2 + 18x 55)

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Solve the problem.56) A private shipping company will accept a box for domestic shipment only if the sum of its

length and girth (distance around) does not exceed 120 in. What dimensions will give a boxwith a square end the largest possible volume?

56)

57) A private shipping company will accept a box for domestic shipment only if the sum of itslength and girth (distance around) does not exceed 120 in. Suppose you want to mail a boxwith square sides so that its dimensions are h by h by w and it's girth is 2h + 2w. Whatdimensions will give the box its largest volume?

57)

58) The strength S of a rectangular wooden beam is proportional to its width times the squareof its depth. Find the dimensions of the strongest beam that can be cut from a11-in.-diameter cylindrical log. (Round answers to the nearest tenth.)

11"

58)

59) The stiffness of a rectangular beam is proportional to its width times the cube of its depth.Find the dimensions of the stiffest beam than can be cut from a 11-in.-diameter cylindricallog. (Round answers to the nearest tenth.)

11"

59)

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60) Suppose that c(x) = 3x3 - 20x2 + 5675x is the cost of manufacturing x items. Find aproduction level that will minimize the average cost of making x items.

60)

61) Suppose c(x) = x3 - 16x2 + 20,000x is the cost of manufacturing x items. Find a productionlevel that will minimize the average cost of making x items.

61)

62) Using the following properties of a twice-differentiable function y = f(x), select a possiblegraph of f.

x y Derivativesx < 2 y > 0,y < 0-2 10 y = 0,y < 0-2 < x < 0 y < 0,y < 00 -6 y < 0,y = 00 < x < 2 y < 0,y > 02 -22 y = 0,y > 0x > 2 y > 0,y > 0

62)

63) Using the following properties of a twice-differentiable function y = f(x), select a possiblegraph of f.

x y Derivativesx < 2 y > 0,y < 0-2 11 y = 0,y < 0-2 < x < 0 y < 0,y < 00 -5 y < 0,y = 00 < x < 2 y < 0,y > 02 -21 y = 0,y > 0x > 2 y > 0,y > 0

63)

Sketch the graph and show all local extrema and inflection points.64) f(x) = -x4 + 4x2 - 4 64)

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65) f(x) = -x4 + 4x2 - 7 65)

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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.66) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of

f.

x y Derivativesx < 2 y > 0,y < 0-2 9 y = 0,y < 0-2 < x < 0 y < 0,y < 00 -7 y < 0,y = 00 < x < 2 y < 0,y > 02 -23 y = 0,y > 0x > 2 y > 0,y > 0

66)

A) B)

C) D)

Sketch the graph and show all local extrema and inflection points.67) f(x) = -x4 + 4x2 - 6 67)

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A) Local maxima: (- 2, -2), ( 2, -2)

Inflection points: -23 ,

29 ,

23 ,

29

B) Local maxima: (- 2, -2), ( 2, -2)Local minimum: (0, -6)No inflection points

C) Local maxima: (- 2, -2), ( 2, -2)Local minimum: (0, -6)

Inflection points: -23 ,

29 ,

23 ,

29

D) Local minima: (- 2, 2), ( 2, 2)Local maximum: (0, 6)

Inflection point: -23 ,

349 ,

23 ,

349

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Answer KeyTestname: TEST3_SAMPLE_PAPER_MTH271

1) Absolute maximum only.2) No absolute extrema.3) Absolute maximum only.4) Absolute minimum and absolute maximum.5) No absolute extrema.6) x = -97) x = -2 and x = 28) x = 0 and x = 69) x = 0, x = -4, and x = 4

10) x = 611) x = 0 and x = 212) x = -2 and x = 213) x = -114) -7, -415) -4, 116) Maximum = 3, 4 ; and minimum = -3, 017) Maximum = (0, 7); minimum = (5, -193)18) absolute maximum is 5 at x = 3; absolute minimum is - 5 at x = -2

19) absolute maximum is 94

at x = 92

; absolute minimum is 0 at 3 and 0 at x = 6

20) absolute maximum is 112

at x = 3; absolute minimum is 3 at x = -2

21) absolute maximum is 14

at x = 72

; absolute minimum is 0 at 4 and 0 at x = 3

22) absolute maximum is 6 at x = 0; absolute minimum is -169 at x = 523) absolute maximum is 6 at x = 0; absolute minimum is -39 at x = 324) x = 325) x = 126) x = 027) x = -228) x = -329) x = -130) No minimum31) No maximum32) Local maximum at (0, 1), local minimum at (2, -3).33) Local maximum at (-2, 18), local minimum at (2, -14).34) None35) Absolute minimum is -4 at x = -1.36) None37) There are no definable extrema.38) (1, )39) (1, )40) (3, )

41) -2 3

3, 2 3

342) Decreasing on (3, 5); increasing on (- , 3) (5, )43) Local minimum at (3, 9)44) Local minimum at (-1, -2)45) Local maximum at (-9, 144)

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Answer KeyTestname: TEST3_SAMPLE_PAPER_MTH271

46) Local maximum at - 17 , -

627

47) Local maximum at (-2, 17); Local minimum at (2, -15)48) Local maximum at (0, -9); Local minima at (1, -10), (-1, -10)49) Concave up for all x50) Concave up on (0, ), concave down on (- , 0)51) Concave up on (- , 0), concave down on (0, )52) Concave up on (-1, ), concave down on (- , -1)53) Concave up on (- , 0), concave down on (0, )54) Concave down on (- , 1), concave up on (1, )55) Concave down on , - 2 , concave up on - 2, 56) 20 in. × 20 in. × 40 in.

57) 803

in. × 803

in. × 20 in.

58) w = 6.4 in.; d = 9.0 in.59) w = 5.5 in.; d = 9.5 in.60) There is not a production level that will minimize average cost.61) 8 items62)

63)

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Answer KeyTestname: TEST3_SAMPLE_PAPER_MTH271

64) Local maxima: (- 2, 0), ( 2, 0)Local minimum: (0, -4)

Inflection points: -23 , 0 ,

23 , 0

65) Local maxima: (- 2, -3), ( 2, -3)Local minimum: (0, -7)

Inflection points: -23 ,

13 ,

23 ,

13

66) C67) C

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