Chapter 6.4 Practice

Questions

1. Take \(f\) to be a differentiable function on \(\mathbb{R}\). A sketch of \(f'\) is given below. Determine all maximal intervals on which \(f\) is increasing and all maximal intervals on which \(f\) is decreasing.

2. Take \(f\) to be a differentiable function on \(\mathbb{R}\). A sketch of \(f'\) is given below. Determine all maximal intervals on which \(f\) is increasing and all maximal intervals on which \(f\) is decreasing.

3. For each function \(f'\) given, determine all maximal intervals on which \(f\) is increasing and which \(f\) is decreasing:
   1. \(f'(x)=x^2(x+5)^8(x+4)^5\)
   2. \(f'(x)=(x+13)^{12}(x+11)^{10}(x+10)^{2}\)
   3. \(f'(x)=(x+13)^3(x+3)^{15}(x+1)^{9}\)
   4. \(f'(x)=e^x-2\)
   5. \(f'(x)=x^3e^x\)
   6. \(f'(x)=|x|-3\)
4. For each function \(f\), determine the intervals for which \(f\) is increasing and intervals for which \(f\) is decreasing.
   1. \(f(x)=x^3-\frac{9}{2}x^2-30x-2\)
   2. \(f(x)=6x^4-8x^3-12x^2+24x-3\)
   3. \(f(x)=\frac{x^2}{x^2+9}\)
   4. \(f(x)=(x-1)^{1/3}(x-4)^{2/3}\)
   5. \(f(x)=\sqrt{x}e^{-x}\)
   6. \(f(x)=-\ln(x^2+1)\)
   7. \(f(x)=\cos(\pi x)\)
5. For each function \(f'\) given below, identify and classify all points at which \(f\) has a local maximum or local minimum.
   1. \(f'(x)=x^2(x+5)^8(x+4)^5\)
   2. \(f'(x)=(x+13)^{12}(x+11)^{10}(x+10)^{2}\)
   3. \(f'(x)=(x+13)^3(x+3)^{15}(x+1)^{9}\)
   4. \(f'(x)=e^x-2\)
   5. \(f'(x)=x^3e^x\)
   6. \(f'(x)=|x|-3\)
6. For each function \(f\), identify and classify all points at which \(f\) has a local maximum or local minimum.
   1. \(f(x)=x^3-\frac{9}{2}x^2-30x-2\)
   2. \(f(x)=6x^4-8x^3-12x^2+24x-3\)
   3. \(f(x)=\frac{x^2}{x^2+9}\)
   4. \(f(x)=(x-1)^{1/3}(x-4)^{2/3}\)
   5. \(f(x)=\sqrt{x}e^{-x}\)
   6. \(f(x)=-\ln(x^2+1)\)
   7. \(f(x)=\cos(\pi x)\)
   8. \(f(x)=x^3-10x^2+x-10\)
   9. \(f(x)=x^3-3x^2+x-3\)
7. Take \(f\) to be a differentiable function on \(\mathbb{R}\). A sketch of \(f'\) is given below. Identify and classify all points at which \(f\) has a local maximum or local minimum.

8. Take \(f\) to be a differentiable function on \(\mathbb{R}\). A sketch of \(f'\) is given below. Identify and classify all points at which \(f\) has a local maximum or local minimum.

9. For each function \(f\) and interval \(I\), identify all critical points and extreme values of \(f\) on \(I.\)

   1. \(f(x)=x^3-3x^2+x-3\); \(I=[0,3]\)
   2. \(f(x)=2x^2+2x+1\); \(I=[-2,2]\)
   3. \(f(x)=\frac{x^3}{(x-2)(x-4)}\); \(I=[3,3.5]\)

10. For each function \(f\), determine \(f''\).

    1. \(f(x)=2x+1\)
    2. \(f(x)=4x^2+5x+1\)
    3. \(f(x)=\frac{x}{x^2+1}\)
    4. \(f(x)=2^{3x+1}-1\)
    5. \(f(x)=\sin(3x+1)+x\)
    6. \(f(x)=(3x+1)\ln(x)\)

11. Take \(f\) to be a function so that \[f''(x)=(x+11)^4(x+10)(x+4)^7.\] Identify all maximal intervals on which \(f\) is convex and concave. Identify all inflection points as well.

12. Take \(f\) to be a function so that \[f''(x)=(x+3)^{12}(x-7)^{14}(x-8)^6.\] Identify all maximal intervals on which \(f\) is convex and concave. Identify all inflection points as well.

13. For each function \(f\), determine the maximal intervals on which \(f\) is convex and concave. Identify all inflection points as well.

    1. \(f(x)=x^3-\frac{9}{2}x^2-30x-2\)
    2. \(f(x)=(x-3)^3\)
    3. \(f(x)=6x^4-8x^3-12x^2+24x-3\)
    4. \(f(x)=xe^{3x}\)
    5. \(f(x)=\ln(x^2+1)\)

14. Take \(f\) to be the function defined on \(\mathbb{R}\) with the following properties. Determine the maximal intervals on which \(f\) is convex and concave. Identify all inflection points as well.

    • \(f''\) is positive on \((-\infty,2)\cup(2,6)\cup(9,10)\)
    • \(f''\) is negative on \((6,9)\cup(10,\infty)\)
    • \(f''\) is zero at \(x=2\), \(x=6\), \(x=9\), or \(x=10\)

Answers

1. Increasing on \((-\infty,3],[4,\infty)\), decreasing on \([3,4]\)

2. Increasing on \([3,4],[5,6]\), decreasing on \((-\infty,3],[4,5],[6,\infty)\)

3. 1. Increasing on \([-4,\infty)\), decreasing on \((-\infty,-4]\)
   2. Increasing on \((-\infty,\infty)\)
   3. Increasing on \([-13,-3],[-1,\infty)\), decreasing on \((-\infty,-13],[-3,-1]\)
   4. Increasing on \([\ln(2),\infty)\), decreasing on \((-\infty,\ln(2)]\)
   5. Increasing on \([0,\infty)\), decreasing on \((-\infty,0]\)
   6. Increasing on \((-\infty,-3],[3,\infty)\), decreasing on \([-3,3]\)
4. 1. Increasing on \((-\infty,-2],[5,\infty)\), decreasing on \((-2,5)\)
   2. Increasing on \([-1,\infty)\), decreasing on \((-\infty,-1]\)
   3. Increasing on \([0,\infty)\), decreasing on \((-\infty,0]\)
   4. Increasing on \((-\infty,2],[4,\infty)\), decreasing on \([2,4]\)
   5. Increasing on \([0,\frac{1}{2}]\), decreasing on \([\frac{1}{2},\infty)\)
   6. Increasing on \((-\infty,0]\), decreasing on \([0,\infty)\)
   7. Increasing on \([2k-1,2k]\) where \(k\) is an integer; decreasing on \([2k,2k+1]\) where \(k\) is an integer
5. 1. Local max \(x=-4\), no local min
   2. No local max or local min
   3. Local min \(x=-13\), \(x=-1\), local max \(x=-3\)
   4. No local max, local min \(x=\ln(2)\)
   5. No local max, local min \(x=0\)
   6. Local max \(x=-3\), local min \(x=3\)
6. 1. Local max \(x=-2\), no local min \(x=5\)
   2. No local max, local min \(x=-1\)
   3. No local max, local min \(x=0\)
   4. Local max \(x=2\), local min \(x=4\)
   5. Local max \(x=\frac{1}{2}\)
   6. Local max \(x=0\)
   7. Local max \(x=2k\), where \(k\) is an integer, local min \(x=2k+1\), where \(k\) is an integer
   8. Local max \(x=\frac{20-2\sqrt{97}}{6}\), local min \(x=\frac{20+2\sqrt{97}}{6}\)
   9. Local max \(x=1-\sqrt{\frac{2}{3}}\), local min \(x=1+\sqrt{\frac{2}{3}}\)

7. Local max \(x=3\), local min \(x=4\)

8. Local max \(x=4\), \(x=5\), local min \(x=3\), \(x=6\)

9. 1. critical values at \(x=1-\sqrt{\frac{2}{3}}\) and \(x=1+\sqrt{\frac{2}{3}}\); local max at \(x=1-\sqrt{\frac{2}{3}}\), local min at \(x=1+\sqrt{\frac{2}{3}}\), global max at \(x=0\), global min at \(x=1+\sqrt{\frac{2}{3}}\)
   2. critical value at \(x=-\frac{1}{2}\);local min \(x=-\frac{1}{2}\), global max \(x=2\), global min \(x=-\frac{1}{2}\)
   3. critical value at \(x=6-\sqrt{12}\); local max at \(x=6-\sqrt{12}\), global max at \(x=6-\sqrt{12}\), global min at \(x=3.5\)
10. 1. \(f''(x)=0\)
    2. \(f''(x)=8\)
    3. \(f''(x)=\frac{2x^3-6x}{(x^2+1)^3}\)
    4. \(f''(x)=9(\ln(2))^22^{3x+1}\)
    5. \(f''(x)=-9\sin(3x+1)\)
    6. \(f''(x)=\frac{3}{x}-\frac{1}{x^2}\)

11. Convex on \((-\infty,-10],[-4,\infty)\), concave on \([-10,-4]\); inflection points at \(x=-10\),\(x=-4\)

12. Convex on \((-\infty,\infty)\); no inflection points

13. 1. Convex on \((\frac{3}{2},\infty)\), concave on \((-\infty,\frac{3}{2})\); inflection point at \(x=\frac{3}{2}\)
    2. Convex on \((3,\infty)\), concave on \((-\infty,3)\); inflection point at \(x=3\)
    3. Convex on \((-\infty,-\frac{1}{3}),(1,\infty)\), concave on \((-\frac{1}{3},1)\); inflection point at \(x=-\frac{1}{3}\), \(x=1\)
    4. Convex on \((-\infty,-\frac{2}{3})\), concave on \((-\infty,-\frac{2}{3})\); inflection point at \(x=-\frac{2}{3}\).
    5. Convex on \((-1,1)\), concave on \((-\infty,-1),(1,\infty)\); inflection at \(x=-1\), \(x=1\).

14. Convex on \((-\infty,2),(2,6),(9,10)\), concave on \((6,9),(10,\infty)\), inflection point at \(x=6, x=9,x=10\)

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