Solutions To Selected Problems Chapter 2 Chapter 4

Chapter 3

Exercises 3.1

1.

Answers will vary.
3.

Answers will vary.
5.

F
7.

$A$ : none; $B$ : abs. max and rel. max; $C$ : rel. min; $D$ : none; $E$ : none; $F$ : rel. min; $G$ : none
9.

$f\,^{\prime}(0)=0$
11.

$f\,^{\prime}(\pi/2)=0$ ; $f\,^{\prime}(3\pi/2)=0$
13.

$f\,^{\prime}(0)=0$
15.

$f\,^{\prime}(2)$ is not defined; $f\,^{\prime}(6)=0$
17.

min: $(-0.5,3.75)$

max: $(2,10)$
19.

min: $(\pi/4,3\sqrt{2}/2)$

max: $(\pi/2,3)$
21.

min: $(\sqrt{3},2\sqrt{3})$

max: $(5,28/5)$
23.

min: $(\pi,-e^{\pi})$

max: $(\pi/4,\frac{\sqrt{2}e^{\pi/4}}{2})$
25.

min: $(1,0)$

max: $(e,1/e)$
27.

Answers will vary.
29.
(a) $x^{3}-x$ , $x^{3}$ , and $x^{3}+x$ have 2, 1, and 0 critical numbers respectively. Because the derivative is a quadratic with at most 2 roots, a cubic cannot have 3 or more critical numbers. (b) A cubic can only have 2 or 0 extreme values.
31.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y(y-2x)}{x(x-2y)}$
33.

$3x^{2}+1$

Exercises 3.2

1.

Answers will vary.
3.

Any $c$ in $(-1,1)$ is valid.
5.

$c=-1/2$
7.

Rolle’s Thm. does not apply.
9.

Rolle’s Thm. does not apply.
11.

$c=0$
13.

$c=3/\sqrt{2}$
15.

The Mean Value Theorem does not apply.
17.

$c=-2/3$
19.

With $c$ given by the Mean Value Theorem, $f(4)=f(1)+f\,^{\prime}(c)(4-1)=10+3f\,^{\prime}(c)\geq 16$ .
21.

$f(-1)<0<f(0)$ , so it has at least one root. $f\,^{\prime}=2+3x^{2}+20x^{4}\geq 2$ , so more than one root would contradict Rolle’s Theorem.
23.

(a) is Rolle’s Theorem. For (b), applying Rolle’s Theorem to roots 1 and 2 and roots 2 and 3 shows that $f\,^{\prime}$ has two roots, and we can then apply (a).
25.

Max value of 19 at $x=-2$ and $x=5$ ; min value of 6.75 at $x=1.5$ .
27.

They are the odd, integer valued multiples of $\pi/2$ (such as $\pm\pi/2,\pm 3\pi/2,\pm 5\pi/2$ , etc.)

Exercises 3.3

1.

Answers will vary.
3.

Answers will vary.
5.

F
7.
decreasing on $[-3,-1]$ ; $[1,3]$ , increasing on $(-\infty,-3]$ ; $[-1,1]$ ; $[3,\infty)$ ; local maxima when $x=-3,1$ , local minima when $x=-1,3$ .
9.
decreasing on $(-\infty,-2]$ ; $[2,\infty)$ , increasing on $[-2,2]$ ; local maxima when $x=2$ , local minima when $x=-2$ .
11.

Graph and verify.
13.

Graph and verify.
15.

Graph and verify.
17.

Graph and verify.
19.
domain: $(-\infty,\infty)$ ; c.p. at $c=-1$ ; decreasing on $(-\infty,-1]$ ; increasing on $[-1,\infty)$ ; rel. min at $x=-1$ .
21.
domain= $(-\infty,\infty)$ ; c.p. at $c=\frac{1}{6}(-1\pm\sqrt{7})$ ; decreasing on $[\frac{1}{6}(-1-\sqrt{7}),\frac{1}{6}(-1+\sqrt{7})]$ ; increasing on $(-\infty,\frac{1}{6}(-1-\sqrt{7})]$ ; $[\frac{1}{6}(-1+\sqrt{7}),\infty)$ ; rel. min at $x=\frac{1}{6}(-1+\sqrt{7})$ ; rel. max at $x=\frac{1}{6}(-1-\sqrt{7})$ .
23.
domain= $(-\infty,\infty)$ ; c.p. at $c=1$ ; decreasing on $[1,\infty)$ increasing on $(-\infty,1]$ ; rel. max at $x=1$ .
25.
domain= $(-\infty,-2)\cup(-2,4)\cup(4,\infty)$ ; no c.p.; decreasing on entire domain, $(-\infty,-2)$ ; $(-2,4)$ ; $(4,\infty)$ .
27.
domain= $(-\pi,\pi)$ ; c.p. at $c=-3\pi/4,-\pi/4,\pi/4,3\pi/4$ ; decreasing on $[-3\pi/4,-\pi/4]$ ; $[\pi/4,3\pi/4]$ ; increasing on $(-\pi,-3\pi/4]$ ; $[-\pi/4,\pi/4]$ ; $[3\pi/4,\pi)$ ; rel. min at $x=-\pi/4,3\pi/4$ ; rel. max at $x=-3\pi/4,\pi/4$ .
29.
domain= $[0,3\pi]$ ; c.p. at $c=\frac{\pi}{3},\frac{5\pi}{3},\frac{7\pi}{3}$ ; decreasing on $[0,\frac{\pi}{3}]$ ; $[\frac{5\pi}{3},\frac{7\pi}{3}]$ ; increasing on $[\frac{\pi}{3},\frac{5\pi}{3}]$ ; $[\frac{7\pi}{3},3\pi]$ ; rel. min at $x=\frac{\pi}{3},\frac{7\pi}{3}$ ; rel. max at $x=\frac{5\pi}{3}$
31.
domain= $[3,\infty)$ ; no c.p.; increasing on $[3,\infty)$
33.
domain= $(-\infty,\infty)$ ; c.p. at $c=-1,0$ ; decreasing on $(-\infty,-1]$ increasing on $[-1,\infty)$ ; rel. min at $x=-1$
35.
domain= $[0,\infty)$ ; c.p. at $c=1/4$ ; decreasing on $[1/4,\infty)$ ; increasing on $[0,1/4]$ ; rel. max at $x=1/4$
37.
domain= $[0,2\pi]$ ; c.p. at $c=0,\pi/2,\pi,3\pi/2,2\pi$ ; decreasing on $[\pi/2,3\pi/2]$ ; increasing on $[0,\pi/2]$ ; $[3\pi/2,2\pi]$ ; rel. max at $x=\pi/2$ ; rel. min at $x=3\pi/2$
39.

Hint/sketch: Suppose that $f\,^{\prime}(c)>0$ and $f\,^{\prime}(d)<0$ . By considering the difference quotient at $x=c$ , explain why the absolute maximum of $f$ on $[c,d]$ cannot occur at $x=c$ . Do the same at $x=d$ . So, by the Extreme Value Theorem (Theorem 3.1.1), $f$ must have a maximum at some $x=r$ in $(c,d)$ , etc.
41.

$c=\pm\cos^{-1}(2/\pi)$

Exercises 3.4

1.

Answers will vary.
3.

Yes; Answers will vary.
5.
concave up on $(-2,2)$ ; concave down on $(-\infty,-2)$ ; $(2,\infty)$ ; inflection points when $x=\pm 2$
7.

Graph and verify.
9.

Graph and verify.
11.

Graph and verify.
13.

Graph and verify.
15.

Graph and verify.
17.
(a) Possible points of inflection: none (b) concave up on $(-\infty,\infty)$ (c) min: $x=1$ (d) $f\,^{\prime}$ has no maximal or minimal value.
19.
(a) Possible points of inflection: $x=0$ (b) concave down on $(-\infty,0)$ ; concave up on $(0,\infty)$ (c) max: $x=-1/\sqrt{3}$ , min: $x=1/\sqrt{3}$ (d) $f\,^{\prime}$ has a minimal value at $x=0$
21.
(a) Possible points of inflection: $x=-2/3,0$ (b) concave down on $(-2/3,0)$ ; concave up on $(-\infty,-2/3)$ , $(0,\infty)$ (c) min: $x=1$ (d) $f\,^{\prime}$ has a relative min at: $x=0$ , relative max at: $x=-2/3$
23.
(a) Possible points of inflection: $x=1$ (b) concave up on $(-\infty,\infty)$ (c) min: $x=1$ (d) $f\,^{\prime}$ has no relative extrema
25.
(a) Possible points of inflection: $x=\pm 1/\sqrt{3}$ (b) concave down on $(-1/\sqrt{3},1/\sqrt{3})$ ; concave up on $(-\infty,-1/\sqrt{3})$ , $(1/\sqrt{3},\infty)$ (c) max: $x=0$ (d) $f\,^{\prime}$ has a relative max at $x=-1/\sqrt{3}$ , relative min at $x=1/\sqrt{3}$
27.
(a) Possible points of inflection: $x=-\pi/4,3\pi/4$ (b) concave down on $(-\pi/4,3\pi/4)$ ; concave up on $(-\pi,-\pi/4)$ , $(3\pi/4,\pi)$ (c) max: $x=\pi/4$ , min: $x=-3\pi/4$ (d) $f\,^{\prime}$ has a relative min at $x=3\pi/4$ , relative max at $x=-\pi/4$
29.
(a) Possible points of inflection: $x=1/e^{3/2}$ (b) concave down on $(0,1/e^{3/2})$ ; concave up on $(1/e^{3/2},\infty)$ (c) min: $x=1/\sqrt{e}$ (d) $f\,^{\prime}$ has a relative min at $x=1/\sqrt{e^{3}}=e^{-3/2}$
31.
(a) Possible points of inflection: none (b) concave up on $(-3,\infty)$ (c) min: $x=-2$ (d) $f\,^{\prime}$ has no relative extrema
33.
(a) Possible points of inflection: $x=1$ (b) concave down on $(-\infty,1)$ ; concave up on $(1,\infty)$ (c) max: $x=-1$ , min: $x=3$ (d) $f\,^{\prime}$ has a relative min at $x=1$
35.
(a) Possible points of inflection: $x=1/2$ (b) concave down on $(1/2,\infty)$ , concave up on $(-\infty,1/2)$ (c) max: $x=1$ , min: $x=0$ (d) $f\,^{\prime}$ has a relative max at $x=1/2$

Exercises 3.5

1.

Answers will vary.
3.

T
5.
concave up on $(-\infty,-1)$ ; $(1,\infty)$ concave down on $(-1,1)$ inflection points when $x=\pm 1$ increasing on $(-2,0)$ ; $(2,\infty)$ decreasing on $(-\infty,-2)$ ; $(0,2)$ relative maximum when $x=0$ relative minima when $x=\pm 2$
7.

A good sketch will include the $x$ and $y$ intercepts and draw the appropriate line.
9.

Use technology to verify sketch.
11.

Use technology to verify sketch.
13.

Use technology to verify sketch.
15.

Use technology to verify sketch.
17.

Use technology to verify sketch.
19.

Use technology to verify sketch.
21.

Use technology to verify sketch.
23.

Use technology to verify sketch.
25.

Use technology to verify sketch.
27.

Use technology to verify sketch.
29.

Use technology to verify sketch.
31.

Use technology to verify sketch.
33.

Use technology to verify sketch.
35.

Use technology to verify sketch.
37.

Use technology to verify sketch.
39.

Use technology to verify sketch.
41.

Use technology to verify sketch.
43.

Use technology to verify sketch.
45.

Use technology to verify sketch.
47.

various possibilities
49.

various possibilities
51.

various possibilities
53.

Critical point: $x=0$ ; Points of inflection: $\pm b/\sqrt{3}$
55.

Critical points: $x=\frac{n\pi/2-b}{a}$ , where $n$ is an odd integer; Points of inflection: $(n\pi-b)/a$ , where $n$ is an integer.
57.

$\dfrac{\operatorname{d}\!y}{\operatorname{d}\!x}=-x/y$ , so the function is increasing in second and fourth quadrants, decreasing in the first and third quadrants.

$\dfrac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=-1/y^{3}$ , which is positive when $y<0$ and is negative when $y>0$ . Hence the function is concave down in the first and second quadrants and concave up in the third and fourth quadrants.