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Question : 19) f(x) = x^2 + 2x - 3 20) f(x) = 2 + 8x - x^2 21) f(x) = x^3 - 3x^2 + 1 : 2151707

Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema.

19) f(x) = x^{2} + 2x - 3

A) Relative minimum of -2 at 0.

B) Relative minimum of 0 at -2.

C) Relative maximum of -4 at -1.

D) Relative minimum of -4 at -1.

20) f(x) = 2 + 8x - x^{2}

A) Relative maximum of 50 at 0.

B) Relative maximum of 16 at - 4.

C) Relative maximum of 18 at 4.

D) Relative minimum of 0 at 4.

21) f(x) = x^{3} - 3x^{2} + 1

A) No relative extrema.

B) Relative maximum of 1 at 0.

C) Relative maximum of 0 at 1; Relative minimum of -3 at -2.

D) Relative maximum of 1 at 0; Relative minimum of -3 at 2.

22) f(x) = x^{3} - 12x + 2

A) Relative maximum of 14 at -2; Relative minimum of 0 at 2.

B) Relative maximum of 18 at -2; Relative minimum of -14 at 2.

C) Relative minimum of -13 at 3.

D) Relative maximum of 1 at 0; Relative minimum of -3 at 2.

23) f(x) = 3x^{4} + 16x^{3} + 24x^{2} + 32

A) Relative minimum of 32 at 0.

B) No relative extrema.

C) Relative maximum of 48 at -2; Relative minimum of 32 at 0.

D) Relative minimum of 30 at -1.

24) f(x) = (x^{2} + 1/x^{2})

A) Relative maximum of 50 at 0.

B) Relative minimum of 0 at 10.

C) Relative maximum of 50 at 0 ; Relative minimum of 0 at 10.

D) No relative extrema.

25) f(x) = (1/x^{2} - 1)

A) No relative extrema.

B) Relative minimum of -1 at 0.

C) Relative maximum of -1 at 0.

D) Relative maximum of 0 at 1.

26) f(x) = x^{2}/5 - 1

A) No relative extrema.

B) Relative minimum of -1 at 0.

C) Relative minimum of -2 at 0.

D) Relative maximum of 2 at 10.

27) f(x) = x^{4}/3 - x^{2}/3

A) Relative minimum of of - (1/4) at (√(2)/4)

B) No relative extrema.

C) Relative maximum of 0 at 0; Relative maximum of - (1/4) at - (√(2)/4)

D) Relative maximum of 0 at 0; Relative minimum of - (1/4) at (√(2)/4) and - (√(2)/4)

28) f(x) = (1/x^{2} + 1)

A) Relative maximum of 0 at 1.

B) Relative minimum of 0.5 at 0.

C) No relative extrema.

D) Relative maximum of 1 at 0.

29) f(x)= (lnx)^{2}, x > 0

A) (1, -1), relative maximum

B) (1, 0), relative minimum

C) (-1, 0), relative minimum

D) (-1, -1) relative maximum

30) f(x) = lnx - x, x > 0

A) (1, 0), relative minimum

B) (-1, 0), relative minimum

C) (1, -1), relative maximum

D) (-1, -1) relative maximum

31) f(x) = x + ln|x|

A) (1, -1), relative maximum

B) (-1, 0), relative minimum

C) (-1, -1) relative maximum

D) (1, 0), relative minimum

32) f(x) = xln|x|, x > 0

A) ((1/e), - (1/e)), relative minimum

B) (- (1/e), - (1/e)), relative maximum

C) ((1/e), (1/e)), relative maximum

D) (- (1/e), (1/e)), relative minimum

33) f(x) = (ln3x)^{2}, x > 0

A) (1, 0), relative minimum

B) ((1/3), 0), relative minimum

C) (3e, 0), relative minimum

D) (-2, 0), relative minimum

34) f(x) = 2xe^{-x}

A) (-1, -2e), relative minimum

B) (-1, -2e), relative maximum

C) (1, (2/e)), relative maximum

D) (1, (2/e)), relative minimum

35) f(x) = xe^{4x}

A) (- (1/4), - (e/4) ), relative maximum

B) ((1/4), - (1/4e) ), relative maximum

C) (- (1/4), - (1/4e) ), relative minimum

D) ((1/4), (e/4) ), relative minimum

36) f(x) = x^{5}e^{x} + 1

A) No relative extrema

B) Relative minimum of -20.06 at -5

C) Relative maximum of 1 at 0; relative minimum of -20.06 at -5

D) Relative maximum of 22.06 at -5; relative minimum of 1 at 0

37) f(x) = (x^{4}/9lnx)

A) Relative minimum of 0 at 0

B) Relative maximum of 0 at 0; relative minimum of (4/9)e at e^{1/4}

C) Relative minimum of (4/9)e at e^{1/4}

D) Relative minimum of - (4/9)e^{-1} at e^{-1/4}

Use the derivative to find the vertex of the parabola.

38) y = 2x^{2} + 12x - 7

A) (3, 25)

B) (3, -25)

C) (-3, 25)

D) (-3, -25)

39) y = -3x^{2} - 18x - 12

A) (-3, 15)

B) (3, 15)

C) (3, -15)

D) (-3, -15)

Use a graphing calculator to find the location of all relative extrema (to three decimal places).

40) f(x) = x^{4} - 3x^{3} - 21x^{2} + 74x - 85

A) Relative maximum at x = 1.604; Relative minima at x = -3.089 and x = 3.735

B) Relative maximum at x = 1.554; Relative minima at x = -3.166 and x = 3.728

C) Relative maximum at x = 1.535; Relative minima at x = -3.111 and x = 3.762

D) Relative maximum at x = 1.69; Relative minima at x = -3.183 and x = 3.637

41) f(x) = x^{4} - 4x^{3} - 53x^{2} - 86x + 11

A) Relative maximum at x = 1.009; relative minima at x = -3.291 and x = 7.226

B) Relative maximum at x = -0.944; relative minima at x = -3.192 and x = 7.136

C) Relative maximum at x = 0.926; relative minima at x = -3.184 and x = 7.122

D) Relative maximum at x = 0.922; relative minima at x = -3.1 and x = 7.187

42) f(x) = x^{5} - 15x^{4} - 3x^{3} - 172x^{2} + 135x + 0.094

A) Relative maximum at x = 0.379; relative minima at x = -0.472 and x = 12.565

B) Relative maximum at x = 0.381; relative minima at x = -0.4and x = -12.638

C) Relative maximum at x= 0.379; relative minimum at x = 12.565

D) Relative maximum at x = 0.319; relative minimum at x = -12.589

Solve the problem.

43) The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately

R(x) = 520x - 0.03x^{2} and C(x) = 120x + 100,000, where x denotes the number of clocks made. What is the maximum annual profit?

A) $1,433,333

B) $1,233,333

C) $1,333,333

D) $1,533,333

44) The annual revenue and cost functions for a manufacturer of precision gauges are approximately

R(x) = 480x - 0.03x^{2} and C(x) = 120x + 100,000, where x denotes the number of gauges made. What is the maximum annual profit?

A) $980,000

B) $1,180,000

C) $1,280,000

D) $1,080,000

45) Find the number of units, x, that produces the maximum profit P, if C(x) = 60 + 48x and

p = 84 - 2x.

A) 36 units

B) 48 units

C) 192 units

D) 9 units

46) Find the maximum profit P if C(x) = 60 + 32x and p = 56 - 2x.

A) $72

B) $988

C) $12

D) $928

47) Find the price p per unit that produces the maximum profit P if C(x) = 35 + 36x and p = 64 - 2x.

A) $46

B) $50

C) $32

D) $28

48) P(x) = -x^{3} + 15x^{2} - 48x + 450, x ≥ 3 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 10 hundred thousand

B) 8 hundred thousand

C) 3 hundred thousand

D) 5 hundred thousand

49) S(x) = -x^{3} + 6x^{2} + 288x + 4000, 4 ≤ x ≤ 20 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon.

A) 8°C

B) 4°C

C) 20°C

D) 12°C