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Solutions

Solutions to Try Its

1. y-intercept (0,0)\left(0,0\right); x-intercepts (0,0),(−5,0),(2,0)\left(0,0\right),\left(-5,0\right),\left(2,0\right), and (3,0)\left(3,0\right) 2. The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity. 3. Graph of f(x)=(1/4)x(x-1)^4(x+3)^3. 4. Because f is a polynomial function and since f(1)f\left(1\right) is negative and f(2)f\left(2\right) is positive, there is at least one real zero between x=1x=1 and x=2x=2. 5. f(x)=−18(x−2)3(x+1)2(x−4)f\left(x\right)=-\frac{1}{8}{\left(x - 2\right)}^{3}{\left(x+1\right)}^{2}\left(x - 4\right) 6. The minimum occurs at approximately the point (0,−6.5)\left(0,-6.5\right), and the maximum occurs at approximately the point (3.5,7)\left(3.5,7\right).

Solutions to Odd-Numbered Exercises

1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which f(x)=0f\left(x\right)=0. 3. If we evaluate the function at a and at b and the sign of the function value changes, then we know a zero exists between a and b. 5. There will be a factor raised to an even power. 7. (−2,0),(3,0),(−5,0)\left(-2,0\right),\left(3,0\right),\left(-5,0\right) 9. (3,0),(−1,0),(0,0)\left(3,0\right),\left(-1,0\right),\left(0,0\right) 11. (0,0), (−5,0), (2,0)\left(0,0\right),\text{ }\left(-5,0\right),\text{ }\left(2,0\right) 13. (0,0), (−5,0), (4,0)\left(0,0\right),\text{ }\left(-5,0\right),\text{ }\left(4,0\right) 15. (2,0), (−2,0), (−1,0)\left(2,0\right),\text{ }\left(-2,0\right),\text{ }\left(-1,0\right) 17. (−2,0),(2,0),(12,0)\left(-2,0\right),\left(2,0\right),\left(\frac{1}{2},0\right) 19. (1,0), (−1,0)\left(1,0\right),\text{ }\left(-1,0\right) 21. (0,0),(3,0),(−3,0)\left(0,0\right),\left(\sqrt{3},0\right),\left(-\sqrt{3},0\right) 23. (0,0), (1,0), (−1,0), (2,0), (−2,0)\left(0,0\right),\text{ }\left(1,0\right)\text{, }\left(-1,0\right),\text{ }\left(2,0\right),\text{ }\left(-2,0\right) 25. f(2)=−10f\left(2\right)=-10 and f(4)=28f\left(4\right)=28. Sign change confirms. 27. f(1)=3f\left(1\right)=3 and f(3)=−77f\left(3\right)=-77. Sign change confirms. 29. f(0.01)=1.000001f\left(0.01\right)=1.000001 and f(0.1)=−7.999f\left(0.1\right)=-7.999. Sign change confirms. 31. 0 with multiplicity 2, −32-\frac{3}{2} with multiplicity 5, 4 with multiplicity 2 33. 0 with multiplicity 2, –2 with multiplicity 2 35. −23 with multiplicity 5,5 with multiplicity 2-\frac{2}{3}\text{ with multiplicity }5\text{,}5\text{ with multiplicity }\text{2} 37. 0 with multiplicity 4,2 with multiplicity 1,−1 with multiplicity 1\text{0}\text{ with multiplicity }4\text{,}2\text{ with multiplicity }1\text{,}-\text{1}\text{ with multiplicity }1 39. 32\frac{3}{2} with multiplicity 2, 0 with multiplicity 3 41. 0 with multiplicity 6,23 with multiplicity 2\text{0}\text{ with multiplicity }6\text{,}\frac{2}{3}\text{ with multiplicity }2 43. x-intercepts, (1,0)\left(1, 0\right) with multiplicity 2, (−4,0)\left(-4, 0\right) with multiplicity 1, y-intercept (0,4)\left(0, 4\right). As x→−∞x\to -\infty , f(x)→−∞f\left(x\right)\to -\infty , as x→∞x\to \infty , f(x)→∞f\left(x\right)\to \infty . Graph of g(x)=(x+4)(x-1)^2. 45. x-intercepts (3,0)\left(3,0\right) with multiplicity 3, (2,0)\left(2,0\right) with multiplicity 2, y-intercept (0,−108)\left(0,-108\right) . As x→−∞x\to -\infty , f(x)→−∞f\left(x\right)\to -\infty , as x→∞x\to \infty , f(x)→∞f\left(x\right)\to \infty . Graph of k(x)=(x-3)^3(x-2)^2. 47. x-intercepts (0,0),(−2,0),(4,0)\left(0, 0\right),\left(-2, 0\right),\left(4, 0\right) with multiplicity 1, y-intercept (0,0)\left(0, 0\right). As x→−∞x\to -\infty , f(x)→∞f\left(x\right)\to \infty , as x→∞x\to \infty , f(x)→−∞f\left(x\right)\to -\infty . Graph of n(x)=-3x(x+2)(x-4). 49. f(x)=−29(x−3)(x+1)(x+3)f\left(x\right)=-\frac{2}{9}\left(x - 3\right)\left(x+1\right)\left(x+3\right) 51. f(x)=14(x+2)2(x−3)f\left(x\right)=\frac{1}{4}{\left(x+2\right)}^{2}\left(x - 3\right) 53. –4, –2, 1, 3 with multiplicity 1 55. –2, 3 each with multiplicity 2 57. f(x)=−23(x+2)(x−1)(x−3)f\left(x\right)=-\frac{2}{3}\left(x+2\right)\left(x - 1\right)\left(x - 3\right) 59. f(x)=13(x−3)2(x−1)2(x+3)f\left(x\right)=\frac{1}{3}{\left(x - 3\right)}^{2}{\left(x - 1\right)}^{2}\left(x+3\right) 61. f(x)=−15(x−1)2(x−3)3f\left(x\right)=-15{\left(x - 1\right)}^{2}{\left(x - 3\right)}^{3} 63. f(x)=−2(x+3)(x+2)(x−1)f\left(x\right)=-2\left(x+3\right)\left(x+2\right)\left(x - 1\right) 65. f(x)=−32(2x−1)2(x−6)(x+2)f\left(x\right)=-\frac{3}{2}{\left(2x - 1\right)}^{2}\left(x - 6\right)\left(x+2\right) 67. local max (−.58, -.62)\left(-\text{.58, -}.62\right), local min (.58, -1.38)\left(\text{.58, -1}\text{.38}\right) 69. global min (−.63, -.47)\left(-\text{.63, -}\text{.47}\right) 71. global min (.75, .89)\text{(}\text{.75, }\text{.89)} 73. f(x)=(x−500)2(x+200)f\left(x\right)={\left(x - 500\right)}^{2}\left(x+200\right) 75. f(x)=4x3−36x2+80xf\left(x\right)=4{x}^{3}-36{x}^{2}+80x 77. f(x)=4x3−36x2+60x+100f\left(x\right)=4{x}^{3}-36{x}^{2}+60x+100 79. f(x)=π(9x3+45x2+72x+36)f\left(x\right)=\pi \left(9{x}^{3}+45{x}^{2}+72x+36\right)

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