3 3826 2 3 4 3 0 2 3 4 3 2 2 3 4 3 f x 1 2 cos x 0 cos x 1 2 f x x 2 sin x x 2

# 3 3826 2 3 4 3 0 2 3 4 3 2 2 3 4 3 f x 1 2 cos x 0

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3 , 3.826 2 3 , 4 3 0, 2 3 , 4 3 , 2 2 3 , 4 3 f x 1 2 cos x 0 cos x 1 2 f x x 2 sin x , 0 < x < 2 Test intervals Sign of Conclusion Increasing Decreasing Increasing f > 0 f < 0 f > 0 f x 4 3 < x < 2 2 3 < x < 4 3 0 < x < 2 3 51. (a) Critical numbers: Increasing on: Decreasing on: (b) Relative maxima: (c) Relative minima: 4 , 0 , 3 4 , 0 , 5 4 , 0 , 7 4 , 0 2 0 1 3 2 , 1 , , 1 , 3 2 , 1 0, 4 , 2 , 3 4 , , 5 4 , 3 2 , 7 4 4 , 2 , 3 4 , , 5 4 , 3 2 , 7 4 , 2 3 2 , 2 , 7 4 , 5 4 , 3 4 , x 4 , f x 2 cos 2 x sin 2 x 0 cos 2 x 0 or sin 2 x 0 f x cos 2 2 x , 0 < x < 2 Test intervals Sign of Conclusion Decreasing Increasing Decreasing Increasing f > 0 f < 0 f > 0 f < 0 f x 3 4 < x < 2 < x < 3 4 4 < x < 2 0 < x < 4 Test intervals Sign of Conclusion Decreasing Increasing Decreasing Increasing f > 0 f < 0 f > 0 f < 0 f x 7 4 < x < 2 3 2 < x < 7 4 5 4 < x < 3 2 < x < 5 4
280 Chapter 4 Applications of Differentiation 52. (a) Critical numbers: Increasing on: Decreasing on: (b) Relative maximum: (c) Relative minimum: 4 3 , 2 2 0 3 3 3 , 2 3 , 4 3 0, 3 , 4 3 , 2 x 3 , 4 3 f x 3 cos x sin x 0 tan x 3 f x 3 sin x cos x , 0 < x < 2 Test intervals Sign of Conclusion Increasing Decreasing Increasing f > 0 f < 0 f > 0 f x 4 3 < x < 2 3 < x < 4 3 0 < x < 3 53. (a) Critical numbers: Increasing on: Decreasing on: (b) Relative minima: Relative maxima: (c) 0 1 2 3 2 , 2 , 3 2 , 0 7 6 , 1 4 , 11 6 , 1 4 2 , 7 6 , 3 2 , 11 6 0, 2 , 7 6 , 3 2 , 11 6 , 2 x 2 , 7 6 , 3 2 , 11 6 f x 2 sin x cos x cos x cos x 2 sin x 1 0 f x sin 2 x sin x , 0 < x < 2 Test intervals Sign of Conclusion Increasing Decreasing Increasing Decreasing Increasing f > 0 f < 0 f > 0 f < 0 f > 0 f x 11 6 < x < 2 3 2 < x < 11 6 7 6 < x < 3 2 2 < x < 7 6 0 < x < 2
Section 4.3 Increasing and Decreasing Functions and the First Derivative Test 281 54. (a) Critical numbers: Increasing on: Decreasing on: (b) Relative maximum: (c) Relative minimum: 3 2 , 1 0 2 2 2 2 , 1 2 , 3 2 0, 2 , 3 2 , 2 x 2 , 3 2 f x cos x 2 sin 2 x 1 cos 2 x 2 0 f x sin x 1 cos 2 x , 0 < x < 2 Test intervals Sign of Conclusion Increasing Decreasing Increasing f > 0 f < 0 f > 0 f x 3 2 < x < 2 2 < x < 3 2 0 < x < 2 55. (a) (b) x 2 1 f 1 2 4 8 10 f 10 8 y f x 2 9 2 x 2 9 x 2 f x 2 x 9 x 2 , 3, 3 (c) Critical numbers: (d) Intervals: Decreasing Increasing Decreasing f is increasing when is positive and decreasing when is negative. f f f x < 0 f x > 0 f x < 0 3 2 2 , 3 3 2 2 , 3 2 2 3, 3 2 2 x ± 3 2 ± 3 2 2 2 9 2 x 2 9 x 2 0 56. (a) (b) x 1 3 4 15 3 6 12 3 1 f f y f x 5 2 x 3 x 2 3 x 16 f x 10 5 x 2 3 x 16 , 0, 5 (c) Critical number: (d) Intervals: Increasing Decreasing f is increasing when is positive and decreasing when is negative. f f f x < 0 f x > 0 3 2 , 5 0, 3 2 x 3 2 5 2 x 3 x 2 3 x 16 0
282 Chapter 4 Applications of Differentiation 57. (a) t t cos t 2 sin t f t t 2 cos t 2 t sin t f t t 2 sin t , 0, 2 (b) t f f 10 20 10 20 30 40 π 2 2 π y (c) Critical numbers: t 2.2889, t 5.0870 t 2.2889, 5.0870 (graphing utility) t cot t 2 t 0 or t 2 tan t t t cos t 2 sin t 0 (d) Intervals: Increasing Decreasing Increasing f is increasing when is positive and decreasing when is negative. f f f t > 0 f t < 0 f t > 0 5.0870, 2 2.2889, 5.0870 0, 2.2889 58. (a) (b) (c) Critical number: (d) Intervals: Increasing Increasing f is increasing when is positive. f f x > 0 f x > 0 , 4 0, x x 2 2 sin x 2 1 1 2 1 2 sin x 2 0 x 8 2 6 4 f f 4 π 3 π 2 π π y f x 1 2 1 2 sin x 2 f x x 2 cos x 2 , 0, 4 59. (a) (b) (c) (d) f > 0 on 2 2 , 3 ; f < 0 on 0, 2 2 f x 0 x 1 2 2 2 x f f 1 1 2 3 4 4 3 2 1 1 y f x 2 x 2 1 2 x f x 1 2