{"id":487,"date":"2021-02-04T15:30:56","date_gmt":"2021-02-04T15:30:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=487"},"modified":"2024-05-14T15:33:37","modified_gmt":"2024-05-14T15:33:37","slug":"problem-set-derivatives-and-the-shape-of-a-graph","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-derivatives-and-the-shape-of-a-graph\/","title":{"raw":"Problem Set: Derivatives and the Shape of a Graph","rendered":"Problem Set: Derivatives and the Shape of a Graph"},"content":{"raw":"
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1.<\/strong> If [latex]c[\/latex] is a critical point of [latex]f(x)[\/latex], when is there no local maximum or minimum at [latex]c[\/latex]? Explain.\u00a0<\/span><\/p>\r\n\r\n<\/div>\r\n

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2.<\/strong> For the function [latex]y=x^3[\/latex], is [latex]x=0[\/latex] both an inflection point and a local maximum\/minimum?<\/p>\r\n[reveal-answer q=\"fs-id1165043308454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043308454\"]\r\n

It is not a local maximum\/minimum because [latex]f^{\\prime}[\/latex] does not change sign<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.<\/strong> For the function [latex]y=x^3[\/latex], is [latex]x=0[\/latex] an inflection point?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.<\/strong> Is it possible for a point [latex]c[\/latex] to be both an inflection point and a local extrema of a twice differentiable function?<\/p>\r\n\r\n

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No<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5.<\/strong> Why do you need continuity for the first derivative test? Come up with an example.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.<\/strong> Explain whether a concave-down function has to cross [latex]y=0[\/latex] for some value of [latex]x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1165043327495\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327495\"]\r\n

False; for example, [latex]y=\\sqrt{x}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.<\/strong> Explain whether a polynomial of degree 2 can have an inflection point.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (8-12), analyze the graphs of [latex]f^{\\prime}[\/latex], then list all intervals where [latex]f[\/latex] is increasing or decreasing.<\/p>\r\n\r\n

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8.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165042476015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042476015\"]Increasing for [latex]-2<x<-1[\/latex] and [latex]x>2[\/latex]; decreasing for [latex]x<-2[\/latex] and [latex]-1<x<2[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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9.\u00a0<\/strong>\"The<\/span><\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>\"The\r\n[reveal-answer q=\"fs-id1165043331180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043331180\"]Decreasing for [latex]x<1[\/latex]; increasing for [latex]x>1[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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11.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
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12.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165043424814\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043424814\"]Decreasing for [latex]-2<x<-1[\/latex] and [latex]1<x<2[\/latex]; increasing for [latex]-1<x<1[\/latex] and [latex]x<-2[\/latex] and [latex]x>2[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n

For the following exercises (13-17), analyze the graphs of [latex]f^{\\prime}[\/latex], then list<\/p>\r\n\r\n

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  1. all intervals where [latex]f[\/latex] is increasing and decreasing and<\/li>\r\n \t
  2. where the minima and maxima are located.<\/li>\r\n<\/ol>\r\n
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    13.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
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    14.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165042367883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042367883\"]\r\n

    a. Increasing over [latex]-2<x<-1, \\, 0<x<1, \\, x>2[\/latex]; decreasing over [latex]x<-2 \\, -1<x<0, \\, 1<x<2[\/latex];<\/p>\r\nb. maxima at [latex]x=-1[\/latex] and [latex]x=1[\/latex], minima at [latex]x=-2[\/latex] and [latex]x=0[\/latex] and [latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    15.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
    16.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165042371995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042371995\"]a. Increasing over [latex]x>0[\/latex], decreasing over [latex]x<0[\/latex];b. Minimum at [latex]x=0[\/latex][\/hidden-answer]<\/div>\r\n
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    17.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n

    For the following exercises (18-22), analyze the graphs of [latex]f^{\\prime}[\/latex], then list all inflection points and intervals [latex]f[\/latex] that are concave up and concave down.<\/p>\r\n\r\n

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    18.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165043312615\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043312615\"]Concave up on all [latex]x[\/latex], no inflection points[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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    19.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
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    20.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165043348466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043348466\"]Concave up on all [latex]x[\/latex], no inflection points[\/hidden-answer]<\/span><\/div>\r\n<\/div>\r\n
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    21.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
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    22.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165042364619\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042364619\"]Concave up for [latex]x<0[\/latex] and [latex]x>1[\/latex], concave down for [latex]0<x<1[\/latex], inflection points at [latex]x=0[\/latex] and [latex]x=1[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n

    For the following exercises (23-27), draw a graph that satisfies the given specifications for the domain [latex]x=[-3,3][\/latex]. The function does not have to be continuous or differentiable.<\/p>\r\n\r\n

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    23.<\/strong> [latex]f(x)>0, \\, f^{\\prime}(x)>0[\/latex] over [latex]x>1, \\, -3<x<0, \\, f^{\\prime}(x)=0[\/latex] over [latex]0<x<1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    24.<\/strong> [latex]f^{\\prime}(x)>0[\/latex] over [latex]x>2, \\, -3<x<-1, \\, f^{\\prime}(x)<0[\/latex] over [latex]-1<x<2, \\, f^{\\prime \\prime}(x)<0[\/latex] for all [latex]x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042705927\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042705927\"]\r\n

    Answers will vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    25.<\/strong> [latex]f^{\\prime \\prime}(x)<0[\/latex] over [latex]-1<x<1, \\, f^{\\prime \\prime}(x)>0, \\, -3<x<-1, \\, 1<x<3[\/latex], local maximum at [latex]x=0[\/latex], local minima at [latex]x=\\pm 2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    26.<\/strong> There is a local maximum at [latex]x=2[\/latex], local minimum at [latex]x=1[\/latex], and the graph is neither concave up nor concave down.<\/p>\r\n[reveal-answer q=\"fs-id1165042418826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042418826\"]\r\n

    Answers will vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    27.<\/strong> There are local maxima at [latex]x=\\pm 1[\/latex], the function is concave up for all [latex]x[\/latex], and the function remains positive for all [latex]x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercise, determine a. intervals where [latex]f[\/latex] is concave up or concave down, and b. the inflection points of [latex]f[\/latex].<\/p>\r\n\r\n

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    30.<\/strong> [latex]f(x)=x^3-4x^2+x+2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042708484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708484\"]\r\n

    a. Concave up for [latex]x>\\frac{4}{3}[\/latex], concave down for [latex]x<\\frac{4}{3}[\/latex] b. Inflection point at [latex]x=\\frac{4}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (31-37), determine<\/p>\r\n\r\n

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    1. intervals where [latex]f[\/latex] is increasing or decreasing,<\/li>\r\n \t
    2. local minima and maxima of [latex]f[\/latex],<\/li>\r\n \t
    3. intervals where [latex]f[\/latex] is concave up and concave down, and<\/li>\r\n \t
    4. the inflection points of [latex]f[\/latex].<\/li>\r\n<\/ol>\r\n
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      31.<\/strong> [latex]f(x)=x^2-6x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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      32.<\/strong> [latex]f(x)=x^3-6x^2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042333208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042333208\"]\r\n

      a. Increasing over [latex]x<0[\/latex] and [latex]x>4[\/latex], decreasing over [latex]0<x<4[\/latex] b. Maximum at [latex]x=0[\/latex], minimum at [latex]x=4[\/latex] c. Concave up for [latex]x>2[\/latex], concave down for [latex]x<2[\/latex] d. Infection point at [latex]x=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      33.<\/strong> [latex]f(x)=x^4-6x^3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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      34.<\/strong> [latex]f(x)=x^{11}-6x^{10}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042711014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711014\"]\r\n

      a. Increasing over [latex]x<0[\/latex] and [latex]x>\\frac{60}{11}[\/latex], decreasing over [latex]0<x<\\frac{60}{11}[\/latex] b. Minimum at [latex]x=\\frac{60}{11}[\/latex] c. Concave down for [latex]x<\\frac{54}{11}[\/latex], concave up for [latex]x>\\frac{54}{11}[\/latex] d. Inflection point at [latex]x=\\frac{54}{11}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      35.<\/strong> [latex]f(x)=x+x^2-x^3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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      36.<\/strong> [latex]f(x)=x^2+x+1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042364262\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042364262\"]\r\n

      a. Increasing over [latex]x>-\\frac{1}{2}[\/latex], decreasing over [latex]x<-\\frac{1}{2}[\/latex] b. Minimum at [latex]x=-\\frac{1}{2}[\/latex] c. Concave up for all [latex]x[\/latex] d. No inflection points<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      37.<\/strong> [latex]f(x)=x^3+x^4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

      For the following exercises (38-47), determine<\/p>\r\n\r\n

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      1. intervals where [latex]f[\/latex] is increasing or decreasing,<\/li>\r\n \t
      2. local minima and maxima of [latex]f[\/latex],<\/li>\r\n \t
      3. intervals where [latex]f[\/latex] is concave up and concave down, and<\/li>\r\n \t
      4. the inflection points of [latex]f[\/latex]. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.<\/li>\r\n<\/ol>\r\n
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        38. [T]\u00a0<\/strong>[latex]f(x)= \\sin (\\pi x)- \\cos (\\pi x)[\/latex] over [latex]x=[-1,1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043327636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327636\"]\r\n

        a. Increases over [latex]-\\frac{1}{4}<x<\\frac{3}{4}[\/latex], decreases over [latex]x>\\frac{3}{4}[\/latex] and [latex]x<-\\frac{1}{4}[\/latex] b. Minimum at [latex]x=-\\frac{1}{4}[\/latex], maximum at [latex]x=\\frac{3}{4}[\/latex] c. Concave up for [latex]-\\frac{3}{4}<x<\\frac{1}{4}[\/latex], concave down for [latex]x<-\\frac{3}{4}[\/latex] and [latex]x>\\frac{1}{4}[\/latex] d. Inflection points at [latex]x=-\\frac{3}{4}, \\, x=\\frac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        39. [T]\u00a0<\/strong>[latex]f(x)=x + \\sin (2x)[\/latex] over [latex]x=\\left[-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        40. [T]\u00a0<\/strong>[latex]f(x)= \\sin x+ \\tan x[\/latex] over [latex]\\left(-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042375682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042375682\"]\r\n

        a. Increasing for all [latex]x[\/latex] b. No local minimum or maximum c. Concave up for [latex]x>0[\/latex], concave down for [latex]x<0[\/latex] d. Inflection point at [latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        41. [T]\u00a0<\/strong>[latex]f(x)=(x-2)^2 (x-4)^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        42. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{1-x}, \\, x \\ne 1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042708183\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708183\"]\r\n

        a. Increasing for all [latex]x[\/latex] where defined b. No local minima or maxima c. Concave up for [latex]x<1[\/latex], concave down for [latex]x>1[\/latex] d. No inflection points in domain<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        43. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{\\sin x}{x}[\/latex] over [latex]x=[-2\\pi ,0) \\cup (0,2\\pi][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        44.<\/strong> [latex]f(x)= \\sin x e^x[\/latex] over [latex]x=[\u2212\\pi ,\\pi][\/latex]<\/p>\r\n\r\n

        \r\n\r\n[reveal-answer q=\"fs-id1165043250995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043250995\"]\r\n

        a. Increasing over [latex]-\\frac{\\pi }{4}<x<\\frac{3\\pi }{4}[\/latex], decreasing over [latex]x>\\frac{3\\pi }{4}, \\, x<-\\frac{\\pi }{4}[\/latex] b. Minimum at [latex]x=-\\frac{\\pi }{4}[\/latex], maximum at [latex]x=\\frac{3\\pi }{4}[\/latex] c. Concave up for [latex]-\\frac{\\pi }{2}<x<\\frac{\\pi }{2}[\/latex], concave down for [latex]x<-\\frac{\\pi }{2}, \\, x>\\frac{\\pi }{2}[\/latex] d. Infection points at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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        45.<\/strong> [latex]f(x)=\\ln x \\sqrt{x}, \\, x>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        46.<\/strong> [latex]f(x)=\\frac{1}{4}\\sqrt{x}+\\frac{1}{x}, \\, x>0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042323564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042323564\"]\r\n

        a. Increasing over [latex]x>4[\/latex], decreasing over [latex]0<x<4[\/latex] b. Minimum at [latex]x=4[\/latex] c. Concave up for [latex]0<x<8\\sqrt[3]{2}[\/latex], concave down for [latex]x>8\\sqrt[3]{2}[\/latex] d. Inflection point at [latex]x=8\\sqrt[3]{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        47.<\/strong> [latex]f(x)=\\dfrac{e^x}{x}, \\, x\\ne 0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

        For the following exercises (48-52), interpret the sentences in terms of [latex]f, \\, f^{\\prime}[\/latex], and [latex]f^{\\prime \\prime}[\/latex].<\/p>\r\n\r\n

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        48.<\/strong> The population is growing more slowly. Here [latex]f[\/latex] is the population.<\/p>\r\n[reveal-answer q=\"fs-id1165042638490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042638490\"]\r\n

        [latex]f>0, \\, f^{\\prime}>0, \\, f^{\\prime \\prime}<0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        49.<\/strong> A bike accelerates faster, but a car goes faster. Here [latex]f[\/latex] represents the Bike\u2019s position minus the Car\u2019s position.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        50.<\/strong> The airplane lands smoothly. Here [latex]f[\/latex] is the plane\u2019s altitude.<\/p>\r\n[reveal-answer q=\"fs-id1165042708328\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708328\"]\r\n

        [latex]f>0, \\, f^{\\prime}<0, \\, f^{\\prime \\prime}<0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        51.<\/strong> Stock prices are at their peak. Here [latex]f[\/latex] is the stock price.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        52.\u00a0<\/strong>The economy is picking up speed. Here [latex]f[\/latex] is a measure of the economy, such as GDP.<\/p>\r\n[reveal-answer q=\"fs-id1165043390851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043390851\"]\r\n

        [latex]f>0, \\, f^{\\prime}>0, \\, f^{\\prime \\prime}>0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        For the following exercises (53-57), consider a third-degree polynomial [latex]f(x)[\/latex], which has the properties [latex]f^{\\prime}(1)=0, \\, f^{\\prime}(3)=0[\/latex]. Determine whether the following statements are true or false<\/em>. Justify your answer.<\/p>\r\n\r\n

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        53.<\/strong> [latex]f(x)=0[\/latex] for some [latex]1 \\le x \\le 3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        54.<\/strong> [latex]f^{\\prime \\prime}(x)=0[\/latex] for some [latex]1 \\le x \\le 3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042707185\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042707185\"]\r\n

        True, by the Mean Value Theorem<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        55.<\/strong> There is no absolute maximum at [latex]x=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        56.<\/strong> If [latex]f(x)[\/latex] has three roots, then it has 1 inflection point.<\/p>\r\n[reveal-answer q=\"fs-id1165043427341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043427341\"]\r\n

        True, examine derivative<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        57.<\/strong> If [latex]f(x)[\/latex] has one inflection point, then it has three real roots.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

        \n

        1.<\/strong> If [latex]c[\/latex] is a critical point of [latex]f(x)[\/latex], when is there no local maximum or minimum at [latex]c[\/latex]? Explain.\u00a0<\/span><\/p>\n<\/div>\n

        \n
        \n

        2.<\/strong> For the function [latex]y=x^3[\/latex], is [latex]x=0[\/latex] both an inflection point and a local maximum\/minimum?<\/p>\n

        Show Solution<\/span><\/p>\n
        \n

        It is not a local maximum\/minimum because [latex]f^{\\prime}[\/latex] does not change sign<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
        \n

        3.<\/strong> For the function [latex]y=x^3[\/latex], is [latex]x=0[\/latex] an inflection point?<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        4.<\/strong> Is it possible for a point [latex]c[\/latex] to be both an inflection point and a local extrema of a twice differentiable function?<\/p>\n

        \n
        Show Solution<\/span><\/p>\n
        \n

        No<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
        \n

        5.<\/strong> Why do you need continuity for the first derivative test? Come up with an example.<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        6.<\/strong> Explain whether a concave-down function has to cross [latex]y=0[\/latex] for some value of [latex]x[\/latex].<\/p>\n

        Show Solution<\/span><\/p>\n
        \n

        False; for example, [latex]y=\\sqrt{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
        \n

        7.<\/strong> Explain whether a polynomial of degree 2 can have an inflection point.<\/p>\n<\/div>\n<\/div>\n

        For the following exercises (8-12), analyze the graphs of [latex]f^{\\prime}[\/latex], then list all intervals where [latex]f[\/latex] is increasing or decreasing.<\/p>\n

        \n
        8.<\/strong>\u00a0\"The<\/p>\n
        Show Solution<\/span><\/p>\n
        Increasing for [latex]-22[\/latex]; decreasing for [latex]x<-2[\/latex] and [latex]-1\n<\/div>\n<\/div>\n<\/div>\n
        \n
        9.\u00a0<\/strong>\"The<\/span><\/div>\n<\/div>\n
        \n
        10.\u00a0<\/strong>\"The<\/p>\n
        Show Solution<\/span><\/p>\n
        Decreasing for [latex]x<1[\/latex]; increasing for [latex]x>1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        11.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
        \n
        12.<\/strong>\u00a0\"The<\/p>\n
        Show Solution<\/span><\/p>\n
        Decreasing for [latex]-22[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        For the following exercises (13-17), analyze the graphs of [latex]f^{\\prime}[\/latex], then list<\/p>\n

          \n
        1. all intervals where [latex]f[\/latex] is increasing and decreasing and<\/li>\n
        2. where the minima and maxima are located.<\/li>\n<\/ol>\n
          \n
          13.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
          \n
          14.<\/strong>\u00a0\"The<\/p>\n
          Show Solution<\/span><\/p>\n
          \n

          a. Increasing over [latex]-22[\/latex]; decreasing over [latex]x<-2 \\, -1\n

          b. maxima at [latex]x=-1[\/latex] and [latex]x=1[\/latex], minima at [latex]x=-2[\/latex] and [latex]x=0[\/latex] and [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          \n
          15.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
          16.<\/strong>\u00a0\"The<\/p>\n
          Show Solution<\/span><\/p>\n
          a. Increasing over [latex]x>0[\/latex], decreasing over [latex]x<0[\/latex];b. Minimum at [latex]x=0[\/latex]<\/div>\n<\/div>\n<\/div>\n
          \n
          17.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n

          For the following exercises (18-22), analyze the graphs of [latex]f^{\\prime}[\/latex], then list all inflection points and intervals [latex]f[\/latex] that are concave up and concave down.<\/p>\n

          \n
          18.<\/strong>\u00a0\"The<\/p>\n
          Show Solution<\/span><\/p>\n
          Concave up on all [latex]x[\/latex], no inflection points<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          19.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
          \n
          20.<\/strong>\u00a0\"The<\/p>\n
          Show Solution<\/span><\/p>\n
          Concave up on all [latex]x[\/latex], no inflection points<\/div>\n<\/div>\n

          <\/span><\/div>\n<\/div>\n

          \n
          21.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
          \n
          22.<\/strong>\u00a0\"The<\/p>\n
          Show Solution<\/span><\/p>\n
          Concave up for [latex]x<0[\/latex] and [latex]x>1[\/latex], concave down for [latex]0\n<\/div>\n<\/div>\n<\/div>\n

          For the following exercises (23-27), draw a graph that satisfies the given specifications for the domain [latex]x=[-3,3][\/latex]. The function does not have to be continuous or differentiable.<\/p>\n

          \n
          \n

          23.<\/strong> [latex]f(x)>0, \\, f^{\\prime}(x)>0[\/latex] over [latex]x>1, \\, -3\n<\/div>\n<\/div>\n

          \n
          \n

          24.<\/strong> [latex]f^{\\prime}(x)>0[\/latex] over [latex]x>2, \\, -3\n

          Show Solution<\/span><\/p>\n
          \n

          Answers will vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          \n
          \n

          25.<\/strong> [latex]f^{\\prime \\prime}(x)<0[\/latex] over [latex]-10, \\, -3\n<\/div>\n<\/div>\n

          \n
          \n

          26.<\/strong> There is a local maximum at [latex]x=2[\/latex], local minimum at [latex]x=1[\/latex], and the graph is neither concave up nor concave down.<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          Answers will vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          \n
          \n

          27.<\/strong> There are local maxima at [latex]x=\\pm 1[\/latex], the function is concave up for all [latex]x[\/latex], and the function remains positive for all [latex]x[\/latex].<\/p>\n<\/div>\n<\/div>\n

          For the following exercise, determine a. intervals where [latex]f[\/latex] is concave up or concave down, and b. the inflection points of [latex]f[\/latex].<\/p>\n

          \n
          \n

          30.<\/strong> [latex]f(x)=x^3-4x^2+x+2[\/latex]<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          a. Concave up for [latex]x>\\frac{4}{3}[\/latex], concave down for [latex]x<\\frac{4}{3}[\/latex] b. Inflection point at [latex]x=\\frac{4}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          For the following exercises (31-37), determine<\/p>\n

            \n
          1. intervals where [latex]f[\/latex] is increasing or decreasing,<\/li>\n
          2. local minima and maxima of [latex]f[\/latex],<\/li>\n
          3. intervals where [latex]f[\/latex] is concave up and concave down, and<\/li>\n
          4. the inflection points of [latex]f[\/latex].<\/li>\n<\/ol>\n
            \n
            \n

            31.<\/strong> [latex]f(x)=x^2-6x[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            32.<\/strong> [latex]f(x)=x^3-6x^2[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. Increasing over [latex]x<0[\/latex] and [latex]x>4[\/latex], decreasing over [latex]02[\/latex], concave down for [latex]x<2[\/latex] d. Infection point at [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            33.<\/strong> [latex]f(x)=x^4-6x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            34.<\/strong> [latex]f(x)=x^{11}-6x^{10}[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. Increasing over [latex]x<0[\/latex] and [latex]x>\\frac{60}{11}[\/latex], decreasing over [latex]0\\frac{54}{11}[\/latex] d. Inflection point at [latex]x=\\frac{54}{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            35.<\/strong> [latex]f(x)=x+x^2-x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            36.<\/strong> [latex]f(x)=x^2+x+1[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. Increasing over [latex]x>-\\frac{1}{2}[\/latex], decreasing over [latex]x<-\\frac{1}{2}[\/latex] b. Minimum at [latex]x=-\\frac{1}{2}[\/latex] c. Concave up for all [latex]x[\/latex] d. No inflection points<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            37.<\/strong> [latex]f(x)=x^3+x^4[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (38-47), determine<\/p>\n

              \n
            1. intervals where [latex]f[\/latex] is increasing or decreasing,<\/li>\n
            2. local minima and maxima of [latex]f[\/latex],<\/li>\n
            3. intervals where [latex]f[\/latex] is concave up and concave down, and<\/li>\n
            4. the inflection points of [latex]f[\/latex]. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.<\/li>\n<\/ol>\n
              \n
              \n

              38. [T]\u00a0<\/strong>[latex]f(x)= \\sin (\\pi x)- \\cos (\\pi x)[\/latex] over [latex]x=[-1,1][\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              a. Increases over [latex]-\\frac{1}{4}\\frac{3}{4}[\/latex] and [latex]x<-\\frac{1}{4}[\/latex] b. Minimum at [latex]x=-\\frac{1}{4}[\/latex], maximum at [latex]x=\\frac{3}{4}[\/latex] c. Concave up for [latex]-\\frac{3}{4}\\frac{1}{4}[\/latex] d. Inflection points at [latex]x=-\\frac{3}{4}, \\, x=\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              39. [T]\u00a0<\/strong>[latex]f(x)=x + \\sin (2x)[\/latex] over [latex]x=\\left[-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              40. [T]\u00a0<\/strong>[latex]f(x)= \\sin x+ \\tan x[\/latex] over [latex]\\left(-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              a. Increasing for all [latex]x[\/latex] b. No local minimum or maximum c. Concave up for [latex]x>0[\/latex], concave down for [latex]x<0[\/latex] d. Inflection point at [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              41. [T]\u00a0<\/strong>[latex]f(x)=(x-2)^2 (x-4)^2[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              42. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{1-x}, \\, x \\ne 1[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              a. Increasing for all [latex]x[\/latex] where defined b. No local minima or maxima c. Concave up for [latex]x<1[\/latex], concave down for [latex]x>1[\/latex] d. No inflection points in domain<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              43. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{\\sin x}{x}[\/latex] over [latex]x=[-2\\pi ,0) \\cup (0,2\\pi][\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              44.<\/strong> [latex]f(x)= \\sin x e^x[\/latex] over [latex]x=[\u2212\\pi ,\\pi][\/latex]<\/p>\n

              \n
              Show Solution<\/span><\/p>\n
              \n

              a. Increasing over [latex]-\\frac{\\pi }{4}\\frac{3\\pi }{4}, \\, x<-\\frac{\\pi }{4}[\/latex] b. Minimum at [latex]x=-\\frac{\\pi }{4}[\/latex], maximum at [latex]x=\\frac{3\\pi }{4}[\/latex] c. Concave up for [latex]-\\frac{\\pi }{2}\\frac{\\pi }{2}[\/latex] d. Infection points at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              45.<\/strong> [latex]f(x)=\\ln x \\sqrt{x}, \\, x>0[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              46.<\/strong> [latex]f(x)=\\frac{1}{4}\\sqrt{x}+\\frac{1}{x}, \\, x>0[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              a. Increasing over [latex]x>4[\/latex], decreasing over [latex]08\\sqrt[3]{2}[\/latex] d. Inflection point at [latex]x=8\\sqrt[3]{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              47.<\/strong> [latex]f(x)=\\dfrac{e^x}{x}, \\, x\\ne 0[\/latex]<\/p>\n<\/div>\n

              For the following exercises (48-52), interpret the sentences in terms of [latex]f, \\, f^{\\prime}[\/latex], and [latex]f^{\\prime \\prime}[\/latex].<\/p>\n

              \n
              \n

              48.<\/strong> The population is growing more slowly. Here [latex]f[\/latex] is the population.<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]f>0, \\, f^{\\prime}>0, \\, f^{\\prime \\prime}<0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              49.<\/strong> A bike accelerates faster, but a car goes faster. Here [latex]f[\/latex] represents the Bike\u2019s position minus the Car\u2019s position.<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              50.<\/strong> The airplane lands smoothly. Here [latex]f[\/latex] is the plane\u2019s altitude.<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]f>0, \\, f^{\\prime}<0, \\, f^{\\prime \\prime}<0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              51.<\/strong> Stock prices are at their peak. Here [latex]f[\/latex] is the stock price.<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              52.\u00a0<\/strong>The economy is picking up speed. Here [latex]f[\/latex] is a measure of the economy, such as GDP.<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]f>0, \\, f^{\\prime}>0, \\, f^{\\prime \\prime}>0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              For the following exercises (53-57), consider a third-degree polynomial [latex]f(x)[\/latex], which has the properties [latex]f^{\\prime}(1)=0, \\, f^{\\prime}(3)=0[\/latex]. Determine whether the following statements are true or false<\/em>. Justify your answer.<\/p>\n

              \n
              \n

              53.<\/strong> [latex]f(x)=0[\/latex] for some [latex]1 \\le x \\le 3[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              54.<\/strong> [latex]f^{\\prime \\prime}(x)=0[\/latex] for some [latex]1 \\le x \\le 3[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              True, by the Mean Value Theorem<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              55.<\/strong> There is no absolute maximum at [latex]x=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              56.<\/strong> If [latex]f(x)[\/latex] has three roots, then it has 1 inflection point.<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              True, examine derivative<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              57.<\/strong> If [latex]f(x)[\/latex] has one inflection point, then it has three real roots.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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