AN22 Answers: Chapter 4 31. ( ) ( )( ) = − + − f x x x x 2 2 Step 1: = − y x3 Step 2: x-intercepts: −2, 0, 2; y-intercept: 0 Step 3: −2, 0, 2: multiplicity 1, crosses Step 4: 20 25 220 5 Step 5: ( ) ( ) − − 1.15, 3.08 ; 1.15, 3.08 Step 6: y 20 x 5 (3, 215) (2, 0) (0, 0) (1.15, 3.08) (22, 0) (23, 15) (21.15, 23.08) Step 7: Range: ( ) −∞∞, Step 8: Increasing on [ ] −1.15, 1.15 Decreasing on ( ] −∞ −, 1.15 and [ )∞ 1.15, 33. ( ) ( ) ( ) = + − f x x x x 4 3 Step 1: = y x3 Step 2: x-intercepts: −4, 0, 3; y-intercept: 0 Step 3: −4, 0, 3: multiplicity 1, crosses Step 4: 32 25 240 5 Step 5: ( ) ( ) − − 2.36, 20.75 ; 1.69, 12.60 Step 6: y 50 x 5 (24, 0) (22.36, 20.75) (25, 240) (1.69, 212.60) (0, 0) (4, 32) (3, 0) Step 7: Range: ( ) −∞∞, Step 8: Increasing on ( ] −∞ −, 2.36 and [ )∞ 1.69, Decreasing on [ ] −2.36, 1.69 35. ( ) ( )( )( ) = + − + f x x x x x 2 6 2 2 Step 1: =y x 2 4 Step 2: x-intercepts: − − 6, 2, 0, 2; y-intercept: 0 Step 3: −6, −2, 0, 2: multiplicity 1, crosses Step 4: 150 27 2225 3 Step 5: ( ) ( ) − − − 4.65, 221.25 ; 1.06, 30.12 ; ( ) − 1.21, 44.25 Step 6: y 700 x 3 (22, 0) (26, 0) (21.06, 30.12) (27, 630) (24.65, 2221.25) (1.21, 244.25) (2, 0) (3, 270) (0, 0) Step 7: Range: [ ) − ∞ 221.25, Step 8: Decreasing on ( ] −∞ −, 4.65 and [ ] −1.06, 1.21 Increasing on [ ] − − 4.65, 1.06 and [ )∞ 1.21, 37. ( ) ( ) ( ) = − + − f x x x x 1 1 2 2 Step 1: = − y x5 Step 2: x-intercepts: −1, 0, 1; y-intercept: 0 Step 3: 1: multiplicity 1, crosses; −1, 0: multiplicity 2, touches Step 4: 1 22 21 2 Step 5: ( ) ( ) ( ) − − 1, 0 ; 0.54, 0.10 ; 0, 0 ; ( ) 0.74, 0.43 Step 6: y 2.5 x 2.5 (20.54, 0.10) (21, 0) (21.5, 1.40625) (1.2, 21.39392) (1, 0) (0, 0) (0.74, 0.43) Step 7: Range: ( ) −∞∞, Step 8: Increasing on [ ] − − 1, 0.54 and [ ] 0, 0.74 Decreasing on ( ] −∞ −, 1, [ ] −0.54, 0 , and [ )∞ 0.74, 39. ( ) ( )( ) = + + f x x x x 5 3 4 4 3 Step 1: = y x 15 5 Step 2: x-intercepts: −4, − 4 3 , 0 y-intercept: 0 Step 3: − − 4, 4 3 : multiplicity 1, crosses 0: multiplicity 3, crosses Step 4: 800 25 2200 2 Step 5: ( ) ( ) − − − 3.30, 742.12 ; 0.97, 15.07 Step 6: y 800 x 3 (24, 0) (23.30, 742.12) (20.97, 215.07) (0, 0) , 0 2 4 3 Step 7: Range: ( ) −∞∞, Step 8: Increasing on ( ] −∞ −, 3.30 and [ ) − ∞ 0.97, Decreasing on [ ] − − 3.30, 0.97 41. ( ) ( )( )( ) = + − − f x x x x 1 5 5 4 5 Step 1: = y x 1 5 3 Step 2: x-intercepts: −5, 4, 5 y-intercept: 20 Step 3: −5, 4, 5: multiplicity 1, crosses Step 4: 30 26 10 25 Step 5: ( ) ( ) − − 1.85, 25.25 ; 4.51, 0.48 Step 6: y x 10 30 (4.51, 20.48) (21.85, 25.25) (25, 0) (4, 0) (0, 20) (5, 0) Step 7: Range: ( ) −∞∞, Step 8: Increasing on ( ] −∞ −, 1.85 and [ )∞ 4.51, Decreasing on [ ] −1.85, 4.51 43. (a)H x 50 40 30 20 10 0 2 4 6 8 10 The relation appears to be cubic. (b) ( ) = − + − H x x x x 0.2096 3.4167 16.5404 1.6667 3 2 (c) 25 (d) 40 10 0 10 (e) 49; this is not resonable.
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