Chapter 7, Answers

Answers to Exercises

Chapter 7

7.1.4

5

(a) \( 1\); (b) \( \frac{\pi}{2} \); (c) \( \sqrt{2}\); (d) \( 1\).

7.1.6

1

1. . Intervals vary. Here are some possibilities (a) \((-2,1)\); (b) \(\left (0, \frac{\pi}{2}\right )\); (c) \(\left (-\frac{1}{2}, \frac{1}{2}\right )\).

2

1. . (a) \(2\sqrt{\frac{7}{3}}\); (b) \(\frac{1}{2}(\sqrt{15} -1)\); (c) \(2-\sqrt 5\).

3

1. . (a) \(f\) is not continuous on the interval; (b) \(f\) is not differentiable on the interval.

4

1. . (a) \(-\frac{1}{4}\); (b) \(\left \{0,\frac{1}{4}\right \}\); (c) \(\left \{\pm \frac{1}{\sqrt{3}}\right \}\); (d) \(\{0, \pm 1\}\); (e) \(\{0, \pm 1\}\); (f) \(2\).

12

1. . \(\ln\frac{\sqrt{3}+2}{\sqrt{2}-1}\).

13

1. . \(\frac{8}{27}(10\sqrt{10}-1)\).

14

1. . \(\frac{59}{24}\).

15

1. . \(\frac{8}{27}(10\sqrt{10}-1)\).

16

1. . \(\frac{3}{2}\).

17

1. . \(\frac{123}{32}\).

18

. (b) \(\frac{1}{2}(\sqrt{2} + \ln (1+\sqrt{2})\).

7.2.2

1

1. . (a) (i) \(\{-1,2\}\); (ii) Increasing on \((-2,-1) \cup (0, 2)\); decreasing on \((-1,0)\cup (2,4)\);

1. \hphantom{\ref{ch07-02-02-q01}.}

1. (b) (i) \(\{-1,1, 3\}\); (ii) increasing on \((-2,-1) \cup (1, 3)\); decreasing on \((-1,1)\cup (3,4)\);

2

1. . (a) \(\{\pm 1\}\); (b) increasing on \((-\infty,-1) \cup (1, \infty)\); decreasing on \((-1,1)\).

3

1. . (a) \(\{0,4\}\); (b) increasing on \((-\infty,0) \cup (4, \infty)\); decreasing on \((0,2)\cup (2,4)\).

4

1. . (a) \(\left \{\pm \frac{1}{e}\right \}\); (b) increasing on \(\left (-\infty, -\frac{1}{e}\right ) \cup \left ( \frac{1}{e},\infty\right )\); decreasing on \(\left ( -\frac{1}{e},0\right )\cup \left (0, \frac{1}{e}\right )\).

5

1. . (a) \(\left \{ \frac{\pi}{2}, \pi, \frac{3\pi}{2}\right \}\); (b) increasing on \(\left (\frac{\pi}{2}, \pi\right ) \cup \left ( \frac{3\pi}{2},2\pi\right )\); decreasing on \(\left (0,\frac{\pi}{2}\right )\cup \left (\pi,\frac{3\pi}{2}\right )\).

6

1. . (a) \(\left \{\frac{1}{2}, \frac{3}{4}\right \}\);

1. (b) increasing on \(\left (0, \frac{1}{2}\right ) \cup \left (\frac{3}{4}, \frac{3}{2}\right )\cup \left (\frac{3}{2}, \infty\right )\); decreasing on \(\left ( \frac{1}{2}, \frac{3}{4}\right )\).

7

1. . (a) \(\left \{\pm \frac{1}{\sqrt 2}\right \}\); (b) increasing on \((-\frac{1}{\sqrt 2},\frac{1}{\sqrt 2})\); decreasing on \(\left (-\infty,-\frac{1}{\sqrt 2}\right )\cup\left (\frac{1}{\sqrt 2},\infty\right )\).

8

1. . (a) \(\left \{0,\pm \frac{\sqrt{3}}{2}\right \}\); (b) increasing on \(\left (-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}\right )\); decreasing on \(\left (-1,-\frac{\sqrt{3}}{2}\right )\cup \left (\frac{\sqrt{3}}{2}, 1\right )\).

9

1. . (a) Hint: \(f'(x) = -8x^3\ln(1-2x^2)\). \(\{0 \}\); (b) increasing on \(\left (-\frac{1}{\sqrt 2},0\right )\); decreasing on \(\left (0,\frac{1}{\sqrt 2}\right )\).

10

1. . (a) \(\left \{0, \pm \frac{2\pi}{3}\right \}\); (b) increasing on \(\left (-\frac{2\pi}{3},0\right ) \cup \left (\frac{2\pi}{3}, \pi \right )\); decreasing on \(\left (-\pi,-\frac{2\pi}{3}\right )\cup \left (0, \frac{2\pi}{3}\right )\).

11

1. . (a) \(\left \{\pm \sqrt{\frac{1}{3}\left (4+\sqrt{67}\right )}\right \}\); (b) \(\left \{\frac{1}{e}\right \}\); (c) \(\left \{0, \pm \sqrt{\frac{3}{2\ln 2}}\right \}\).

12. \(b=-8\), \(c=7\).
13. (a) \(-\frac{1}{4} \ln\left (\sin 2x\right ) -3\), \(x\in \left (0, \frac{\pi}{2}\right )\);
(b) \(\frac{1}{\pi^2}\ln\left |\sec \pi x\right | -\frac{1}{2}x^2 + x + \frac{7}{2}\), \(x\in \left (\frac{1}{2}, \frac{3}{2}\right )\).
14. (a) Hint: \(f'(x) = \cos 3x +\cos x = \cos (2x+x) +\cos x\) use the addition identity, and double-
\hphantom{\ref{ch07-02-02-q14}.}
angle identities. \(x\in \left \{\frac{\pi}{4}+\frac{\pi}{2}k\; \text{ or } \; \frac{\pi}{2}+ \pi k\, : \, k \in \mathbb{Z}\right \}\).
(b) \(x\in \left \{\frac{\pi}{2}k\,: \, k \in \mathbb{Z}\right \}\).
15.
(a) \(\frac{1}{4}x^2-\frac{\pi }{8} x+\frac{1}{32} \cos 4 x +\frac{\pi ^2}{64}-\frac{95}{32}\); (b) \(\frac{1}{4}x^2-\frac{1}{4} x-\frac{1}{8 \pi ^2}\cos 2 \pi x – \frac{1}{8 \pi ^2}+\frac{65}{16}\).

7.2.4

1

1. . (a) Rel.\! max.: \(\{f(t), f(v)\}\); rel.\! min.: \(\{f(r), f(u)\}\);

1. \hphantom{\ref{ch07-02-04-q01}.}

1. (b) rel.\! max.: \(\{ f(-1), f(3)\}\); rel.\! min.: \(\{f(-2), f(1), f(4)\}\).

2

1. . Rel.\! max.: \(f(-1)=2\); rel.\! min.: \(f(1) = -2\).

3

1. . Rel.\! max.: \(f( 0)= 0\); rel.\! min.: \(f(4 ) = 8\).

4

1. . Rel.\! max.: \(f\left (-\frac{1}{e} \right )= \frac{2}{e}\); rel.\! min.: \(f\left ( \frac{1}{e}\right ) = -\frac{2}{e}\).

5

1. . Rel.\! max.: \(f\left (0 \right )=2 \), \(f\left (\pi \right )=2 \), and \(f\left (2\pi \right )=2 \); rel.\! min.: \(f\left (\frac{\pi}{2} \right ) = -1\) and \(f\left (\frac{3\pi}{2} \right ) = -1\).

6

1. . Rel.\! max.: \(f\left ( \frac{1}{2}\right )= \frac{1}{2\sqrt{2}e}\); rel.\! min.: \(f\left ( \frac{3}{4}\right ) = \frac{1}{e\,\sqrt{3e}}\).

7

1. . Rel.\! max.: \(f\left (\frac{1}{\sqrt 2} \right )=\arctan \frac{1}{\sqrt{2e}} \); rel.\! min.: \(f\left (-\frac{1}{\sqrt 2} \right )=-\arctan \frac{1}{\sqrt{2e}} \).

8

1. . Rel.\! max.: \(f\left (\frac{\sqrt{3}}{2} \right )= 1\); rel.\! min.: \(f\left (-\frac{\sqrt{3}}{2} \right ) = -1\). Note that \(x=0\) is a critical number, but

1. \hphantom{\ref{ch07-02-04-q08}.}

1. is neither a relative maximum nor a relative minimum.

9

1. . Rel.\! max.: \(f\left ( 0\right )= 0\); rel.\! min.: none.

10

1. . Rel.\! max.: \(f\left (-\pi \right )= -\frac{1}{2}=f(\pi)\), and \(f(0)=\frac{3}{2}\); rel.\! min.: \(f\left (-\frac{2\pi}{3} \right ) = -1= f\left (\frac{2\pi}{3} \right ) \).

11

1. . (a) None;

1. (b) incr.\! on \( \left ( -\infty, 0 \right ) \cup \left (0,-\infty \right ) \), decr. nowhere;

1. (c) extreme values none.

12

1. . (a) \(\left \{ -\frac{11\pi}{12}, -\frac{7\pi}{12}, \frac{\pi}{12}, \frac{5\pi}{12}\right \}\);

1. \hphantom{\ref{ch07-02-04-q12}.}

1. (b) incr.\! on \( \left (-\pi, -\frac{11\pi}{12} \right ) \cup \left ( -\frac{7\pi}{12}, \frac{\pi}{12} \right ) \cup \left ( \frac{5\pi}{12}, \pi \right ) \), decr.\! on \( \left (-\frac{11\pi}{12},-\frac{7\pi}{12} \right ) \cup \left ( \frac{\pi}{12}, \frac{5\pi}{12} \right )\);

1. \hphantom{\ref{ch07-02-04-q12}.}

1. (c) local max.\! values:\!\! \(\left \{ f\left ( -\frac{11\pi}{12}\right ), f\left ( \frac{\pi}{12}\right ) , f\left (\pi\right ) \right \}\), local min.\! values:\!\! \(\left \{f\left ( -\pi\right ), f\left ( -\frac{7\pi}{12}\right ), f\left ( \frac{5\pi}{12}\right ) \right \}\).

13

1. . (a) \(\left \{\pm\frac{1}{2\sqrt{2\ln 2}} \right \}\);

1. (b) incr.\! on \( \left ( -\frac{1}{2\sqrt{2\ln 2}} ,\frac{1}{2\sqrt{2\ln 2}} \right ) \), decr.\! on \( \left (-\infty, -\frac{1}{2\sqrt{2\ln 2}} \right )\cup \left (\frac{1}{2\sqrt{2\ln 2}},\infty \right ) \);

1. \hphantom{\ref{ch07-02-04-q13}.}

1. (c) local max.\! values: \(\left \{ f\left (\frac{1}{2\sqrt{2\ln 2}} \right ) \right \}\), local min.\! values: \(\left \{ f\left (-\frac{1}{2\sqrt{2\ln 2}} \right ) \right \}\).

14

1. }. (a) \(\left \{ \pm \frac{1}{2} \right \}\);

1. (b) incr.\! on \( \left ( -\frac{1}{2} ,\frac{1}{2}\right ) \), decr.\! on \( \left (-\infty, -\frac{1}{2}\right )\cup \left (\frac{1}{2},\infty \right ) \);

1. \hphantom{\ref{ch07-02-04-q14}.}

1. (c) local max.\! values: \(\left \{ f\left (\frac{1}{2}\right ) \right \}\), local min.\! values: \(\left \{ f\left (-\frac{1}{2}\right ) \right \}\).

15

1. . (a) \(\left \{\frac{\pi}{2}k\,:\,k \in \mathbb{Z} \right \}\);

1. (b) incr.\! on \( \left (\pi k, \frac{\pi}{2}+ \pi k \right ) \), \(k \in \mathbb{Z}\), decr.\! on \( \left ( \frac{\pi}{2}+ \pi k, \pi + \pi k \right ) \), \(k \in \mathbb{Z}\);

1. \hphantom{\ref{ch07-02-04-q15}.}

1. (c) local max.\! values: \(\left \{ f\left (\frac{\pi}{2}+ \pi k \right ) \right \}\), \(k \in \mathbb{Z}\), local min.\! values: \(\left \{ f\left ( \pi k \right ) \right \}\), \(k \in \mathbb{Z}\).

16

1. . (a) \(\left \{ 0 \right \}\);

1. (b) incr.\! on \( \left ( 0, \infty \right ) \), decr.\! on \( \left (-\infty, 0 \right ) \);

1. \hphantom{\ref{ch07-02-04-q16}.}

(c) local max.\! values: none, local min.\! values: \(\left \{ f\left (0\right ) \right \}\).

7.2.6

1. Abs. max. value: \(1 \); abs.min.value: \(0 \).
2. Abs. max. value: \(4\ln^28 \); abs.min.value: \( 0\).
3. Abs. max. value: none; abs.min.value: \( 18\sqrt[3]{2}\).
4. Abs. max.value: \(\frac{\pi}{4} \); abs.min.value: \(-\frac{\pi}{4} \).
5. Abs. max.value: \(2^{1/3}\sqrt{3}5^{5/6} \); abs.min.value: \(-2^{1/3}\sqrt{3}5^{5/6} \).
6. Abs. max.value: none; abs.min.value: none.
7. Abs. max. value: \(\frac{\pi}{6} \); abs.min.value: \(0\).
8. Abs. max.value: none; abs.min.value: \(5\sqrt{5} \).
9. Abs. max.value: \(1 \); abs.min.value: \(e^{-3\pi/4}/\sqrt{2} \).
10. Abs. max.value: none; abs.min.value: \(2\sqrt{2} \).
11. (a) \(\frac{1}{2}+\frac{1}{2} = 1\); (b) \(\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4} \).
12. Square \(5\times 5 \) m; abs.max. area \(25 \) m\(^2\).
13. \(\left (\frac{1}{10}, -\frac{1}{5}\right ) \).
14. Abs. max. area: only a circle \(\frac{9}{4} \) m\(^2\); abs.min. area:\(\frac{9}{11}(3\sqrt{3}-4) \) m\(^2\), cut at \(\frac{24\sqrt{3}}{9+4\sqrt{3}}\).
15}. Abs. max. value: \(8^3=512\); abs.min. value: \(6^2+2^3 = 44\).
16. \(8-4=4\), abs.max. value\(4\cdot\frac{1}{8^2}= \); abs.min. value: none.
17. \(\left (\frac{1}{2}, \frac{1}{\sqrt{2}}\right ) \).
18. (a) Max. area: \(\frac{32}{3\sqrt{3}}\), base: \(\frac{4}{\sqrt{3}}\), height: \(\frac{4\left (2\sqrt{3}-1\right )}{3}\);
(b) Max. area: \(\frac{\sqrt{2e}}{e}\), base: \(\sqrt{2}\), height: \(\frac{1}{\sqrt{e}}\).
19.
20. In general, there is a function for which it is necessary to find its absolute maximum value
\hphantom{\ref{ch07-02-06-q20}.}
or its absolute minimum value, and there is an equation which is used to eliminate one of
\hphantom{\ref{ch07-02-06-q20}.}
the variables.

7.3.2

1

1. . \(150\) cm\(^2\).

2

1. . Max. area, a circle: \(A= 16\pi\) m\(^2\); min. area: \(\frac{4}{4+\pi}\) m\(^2\), use \(\frac{16}{4+\pi}\) m for the square.

3

1. . \(30\,000\) m\(^2\).

4

1. . \(12\sqrt[3]{2^2} \) m\(^2\).

5

1. . \(\frac{16}{9}\sqrt{6} \) m\(^3\).

6

1. . Square, \(l=w= \sqrt{2} R\), \(A= 2r^2\).

7

1. . \(\frac{10\sqrt{6}}{3} \) m\(^3\).

8

1. . Length of the sides \(\frac{20}{3}\) cm.

9

1. . \(A = 3\sqrt{3}\) m\(^2\).

10

1. . \(10-\sqrt{2} \) km away from \(A\).

11

1. . \(\frac{10}{3}\) m away from the short pole; (b) \(\sqrt{181}\) m.

12

1. . \(48\sqrt{3} \) cm\(^2\).

13

1. . \(13\sqrt{13} \) ft.

14

1. . (a) Walk around the pool; (b) \(\alpha = \frac{\pi}{6}\).

15

1. . \(b=\sqrt{3} R\) m, \( h = \frac{3R}{2}\) m, max. area \(=\frac{3\sqrt{3}}R^2{4}\) m\(^2\).

16

. (a) \(\pi R^2\left (1+\sqrt{5}\right ) \); (b) \(\frac{4\pi R^3}{3\sqrt{3}} \).

7.4.2

1

1. . (a) (i) \(\{-1,0, 2\}\); (ii) concave upward on \((-2,-1) \cup (0, 2)\); concave downward on \((-1,0)\cup (2,4)\);

1. \hphantom{\ref{ch07-04-02-q01}.}

1. (b) (i) \(\{-1,1, 3\}\); (ii) concave upward on \((-2,-1) \cup (1, 3)\); concave downward on \((-1,1)\cup (3,4)\);

2

1. . (a) \(\{0\}\); (b) concave upward on \((0, \infty)\); concave downward on \((-\infty,0)\).

3

1. . (a) None; (b) concave upward on \( (2, \infty)\); concave downward on \((-\infty,2)\).

4

1. . (a) None; (b) concave upward on \(\left (0,\infty\right )\); concave downward on \(\left ( -\infty,0\right )\).

5

1. . (a) \(\left \{ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\right \}\); (b) con. up on \(\left (\frac{\pi}{4}, \frac{\pi}{4}\right ) \cup \left ( \frac{5\pi}{4},\frac{7\pi}{4}\right )\); con. down on \(\left (0,\frac{\pi}{4}\right )\cup \left (\frac{3\pi}{4},\frac{5\pi}{4}\right )\cup \left (\frac{7\pi}{4},2\pi \right )\).

6

1. . (a) \(\left \{\frac{1}{2}\left (2\sqrt{3}-2\right )\right \}\);

1. (b) con. up on \(\left ( \frac{1}{2}\left (2\sqrt{3}-2\right ), \frac{3}{2}\right )\); con. down on \(\left (0, \frac{1}{2}\left (2\sqrt{3}-2\right )\right ) \cup \left (\frac{3}{2}, \infty\right )\).

7

1. . (a) \(\left \{-1\right \}\); (b) concave upward on \((-\infty,-1)\); concave downward on \( \left (-1,\infty\right )\).

8

1. . (a) \(\left \{ \frac{1}{\sqrt{e}}\right \}\); (b) concave upward on \(\left (0, \frac{1}{\sqrt{e}}\right )\); concave downward on \(\left ( \frac{1}{\sqrt{e}},\infty\right )\).

9

1. . (a) \(\{\pm \frac{1}{2\sqrt{2\ln 2}} \}\); (b) con. up \(\left (-\frac{1}{2\sqrt{2\ln 2}},\frac{1}{2\sqrt{2\ln 2}}\right )\); con. down \(\left (-\infty, -\frac{1}{2\sqrt{2\ln 2}}\right )\cup\left (\frac{1}{2\sqrt{2\ln 2}},\infty\right )\).

10

1. . (a) \(\left \{4\right \}\); (b) concave upward on \(\left (4, \infty\right )\); concave downward on \(\left (-\infty, 1\right ) \cup \left (1, 4 \right )\).

11

1. . (a) \(\left \{\pm \sqrt{\frac{1}{3}\left (4+\sqrt{67}\right )}\right \}\); (b) \(\left \{\frac{1}{e}\right \}\); (c) \(\left \{0, \pm \sqrt{\frac{3}{2\ln 2}}\right \}\).

12. \(a=6\), \(b=-18\), \(c=13\).

7.4.4

1

1. (a) \(f'(x)=2 (x+3)^2 (2 x+9)\), \(f”(x) = 12 (x^2+7 x+12)\);

1. (b) local max. none, local min. \(f\left ( -\frac{9}{2}\right ) = -\frac{27}{16}\);

1. (c) incr. on\(\left (-\frac{19}{4} ,\infty \right )\), decr. on \(\left (-\infty ,-\frac{19}{4}\right )\);

1. (d) concave up \(\left (-\infty ,-4\right )\cup \left (-3, \infty \right )\), down \(\left (-4 ,-3\right )\); infl. pt. \(\left \{\left (-4, -1 \right ), \left (-3,0\right )\right \}\);

1. (e) see graph.

2

1. (a) \(f'(x)=4 (x-5) (x-2) (x+1) \), \(f”(x) = 12 (x^2-4 x+1) \);

1. (b) local max. \(f\left (2 \right ) =81 \), local min. \(f\left (-1 \right ) =0 \), \(f\left (5 \right ) =0 \);

1. (c) incr. on \(\left (-1 ,2\right )\cup\left (5,\infty\right )\), decr. on \(\left (-\infty ,-1\right ) \cup \left (2 ,5\right )\);

1. (d) concave up \(\left ( -\infty, 2-\sqrt{3}\right )\cup \left (2+\sqrt{3}, \infty\right )\), down \(\left ( 2-\sqrt{3}, 2+\sqrt{3}\right )\); infl. pt. \(\left \{ \left (2\pm\sqrt{3}, 36 \right ) \right \}\);

1. (e) see graph.

3

1. (a) \(f'(x)= -\frac{x+5}{(x+4)^3}\), \(f”(x) =\frac{2(x+7)}{(x+4)^4} \);

1. (b) local max. none, local min. \(f\left (-6 \right ) = -\frac{1}{4}\);

1. (c) incr. on \(\left (-6,4\right )\), decr. on \(\left (-\infty ,-6\right )\cup \left (4,\infty\right )\);

1. (d) concave up \(\left (-7,-4\right ) \cup (4, \infty) \), down \(\left (-\infty ,-7\right )\); infl. pt. \( \left \{ \left (-7 , -\frac{2}{9} \right ) \right \}\);

1. (e) see graph.

4

1. (a) \(f'(x)= \frac{2(x+2)}{(x+3)^3}\), \(f”(x) = -\frac{2(2x+3)}{(x+3)^4} \);

1. (b) local max. none, local min. \(f\left ( -2\right ) = 0 \);

1. (c) incr. on \(\left (-\infty , -3\right )\cup (-2,\infty)\), decr. on \(\left ( -3,-2\right )\);

1. (d) concave up \(\left (-\infty , -3\right )\cup\left (-3, -\frac{3}{2}\right )\), down \(\left (-\frac{3}{2} , \infty\right )\); infl. pt. \(\left \{ \left (-\frac{3}{2}, \frac{1}{9} \right ) \right \}\);

1. (e) see graph.

1. \end{itemize}

5. Abs. min. \(f\left (\sqrt{2}\right ) =4 \).
6. Abs. max. \(f\left (-3\right ) =-6 \).
7. Abs, min. \(f\left (3\right ) =\frac{3}{4} \).
8. None.
9. (a) local max. \(f\left (\frac{1}{2}\right ) = \frac{1}{4e^2}\); local min. \(f\left (0\right ) = 0\); (b) \( x\in \left \{\frac{1}{4}\left (2\pm \sqrt{2}\right )\right \}\).
10. (a) local max. \(f\left (3\right ) = \frac{9}{e^3}\); local min. \(f\left (\frac{1}{2}\right ) = -\frac{1}{\sqrt{e}}\); (b) \( x\in \left \{\frac{1}{4}\left (11\pm \sqrt{41}\right )\right \}\).
11. (a) None; (b) \(x\in \left \{\pm \sqrt{\frac{2}{3}}\right \}\).
12. (a) local max. \(f\left (-\frac{1}{e}\right ) = \frac{2}{e}\); local min. \(f\left (\frac{1}{e}\right ) = -\frac{2}{e}\); (b) none.
15. \(f”(x)=K(x + 2) (x – 1) \) for infl. pts. at \(x=-2,1\). \(\Longrightarrow f(x) = K\left (\frac{1}{12} x^4 +\frac{1}{6}x^3-x^2\right ) + c_1 x + c_2\).