Domain: ( −∞ ,0 )( ∪ 0, ∞ )

16. Domain: ( −∞ ,0 )( ∪ 0, ∞ )

Range: ( −∞ − ∪ , 2] [2, ) ∞

Odd function since

2 2 18. Domain: (– ( ∞ , ∞ ); range [0, ∞ ) −

w ( −= z )

3 f 3 (– ) x = – x = x = fx ( ); even function; − z

=− wz () ; symmetric

with respect to the origin.

symmetric with respect to the y-axis.

y-intercept: none

y -intercept: 0; x-intercept: 0

x-intercept: none

1 fx () = 3 x ⎜ ⎟ = 3 xx ; () fx ′ = 0 when x = 0

wz '( ) =− 1 2 ; wz '( ) = 0 when z =± 1 .

critical points: z =± 1 . wz '( ) > 0 on

Critical point: 0

fx ′ () > 0 when x > 0

( −∞ − ∪ ∞ , 1) (1, ) so the function is increasing on

f (x) is increasing on [0, ∞ ) and decreasing on ( −∞ − ∪ ∞ , 1] [1, ) . The function is decreasing on

[ 1, 0) − ∪ (0,1) . Global minimum f(0) = 0; no local maxima

3 x local minimum 2 w (1) = 2 and local maximum

f 2 ′′ 2 () x = 3 x + = 6 x as x = x ;

w ( 1) −=− 2 . No global extrema.

f ′′ () x > 0 when x ≠ 0

f (x) is concave up on (– ∞ , 0) ∪ (0, ∞ ); no z

wz ''( ) = 3 > 0 when z > 0 . Concave up on

(0, ) ∞ and concave down on ( −∞ ,0 ) .

inflection points

No horizontal asymptote; x = 0 is a vertical

asymptote; the line y = z is an oblique (or slant) asymptote.

No inflection points.

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19. Domain: (– ∞ , ∞ ); range: (– ∞ , ∞ )

21. Domain: (– ∞ , ∞ ); range: [0, ∞ )

Neither an even nor an odd function. symmetric with respect to the origin.

R (– ) z = –– z z = – zz = – ( ); Rz odd function;

Note that for x ≤ 0, x = – x so x + = while x 0, y -intercept: 0; z-intercept: 0

for x > 0, x = so x

Rz 2 ′ () = z + = 2 z since z 2 = z for all z;

() ⎧⎪ 0 if x ≤ gx 0

Rz ′ () = 0 when z = 0

3 ⎪⎩ 2 x + 2 x if x > 0

Critical point: 0 y -intercept: 0; x-intercepts: ( −∞ , 0]

Rz ′ () > 0 when z ≠ 0

R (z) is increasing on (– ∞ ,

∞ ) and decreasing

No local minima or maxima

No critical points for x > 0.

g (x) is increasing on [0, ∞ ) and decreasing

Rz ′′ () = ; () Rz ′′ > 0 when z > 0.

nowhere.

⎧ 0 if 0 x ≤

R (z) is concave up on (0, ∞ ) and concave down

gx ′′ () =⎨

on (– ∞ , 0); inflection point (0, 0).

⎩ 6 if 0 x >

g (x) is concave up on (0, ∞ ); no inflection points

20. Domain: (– ∞ , ∞ ); range: [0, ∞ )

H (– ) q = (– ) – q 2 q = qq 2 = Hq ( ); even

22. Domain: (– ∞ , ∞ ); range: [0, ∞ ) Neither an even nor an odd function. Note that

function; symmetric with respect to the y-axis. y -intercept: 0; q-intercept: 0

for x < 0, x = – x so

= – x , while for

for all q; Hq () = when q = 0 0 ⎧− + − ⎪ x Critical point: 0 3 x 2 6 x if x < 0

hx () =⎨

Hq ′ () > when q > 0 0 ⎪⎩ 0 if x ≥ 0

y -intercept: 0; x-intercepts: [0, ∞ ) decreasing on (– ∞ , 0].

H (q) is increasing on [0, ∞ ) and

⎧− ⎪ 3 2 2 − 6 if < 0 Global minimum H(0) = 0; no local maxima

= 6 qHq ; () ′′ > when 0 No critical points for x < 0

h (x) is increasing nowhere and decreasing on

q ≠ 0.

H (q) is concave up on (– ∞ , 0) ∪ (0, ∞ ); no

⎧ −+ 6 x 2 if x < 0

inflection points.

hx ′′ () =⎨

⎩ 0 if x ≥ 0

h (x) is concave up on (– ∞ , 0); no inflection

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24. Domain: [2k π , (2k + 1)π ] where k is any integer; range: [0, 1] Neither an even nor an odd function y -intercept: 0; x-intercepts: k π , where k is any integer.

fx ′ () =

cos x

; () fx ′ = when 0 x = 2 k π+

2 while fx ′ () does not exist when x = k π , k any integer.

2 sin x

Critical points: k π ,2 k π + where k is any

23. Domain: (– ∞ , ∞ ); range: [0, 1]

f ( −= x ) sin( − x ) =− sin x = sin x = fx ( ); even

integer

function; symmetric with respect to the y-axis.

fx ′ () > when 2 0 k π<< x 2 k π+

y -intercept: 0; x-intercepts: k π where k is any

integer.

(x) is increasing on 2 ⎡ , 2

= when 0 x =+π k

⎢ k π+ , (2 k +π 1) ⎥ , k any and fx ′ () does not exist when x = k π , where k

sin x

decreasing on 2

is any integer.

integer.

Global minima f(k π ) = 0; global maxima Critical points:

π k π + , where k is any

and

f ⎜ 2 π+ k ⎞ ⎟ = 1, k any integer

integer; fx ′ () > 0 when sin x and cos x are either

both positive or both negative. 2 – cos x – 2sin 2 x –1 – sin 2 x

f ′′ () x =

f (x) is increasing on ⎡ ⎢ k ππ+ , k ⎤ ⎥ and decreasing

4sin

4sin x

+ 1 sin 2 x

3/2 ; ⎡ π on ⎢ k π+ ,( k +π 1) ⎤ ⎥ where k is any integer.

4 sin

f ′′ () x < for all x. 0

Global minima f(k π ) = 0; global maxima

f (x) is concave down on (2k π , (2k + 1)π ); ⎛

π fk π+ ⎞ = 1, where k is any integer. ⎜ no inflection points 2 ⎝ ⎟ ⎠

⎟ sin x ⎟ (cos ) x

f ′′ () x < when x 0 ≠ k π , k any integer

25. Domain: ( −∞ ∞ ,)

f (x) is never concave up and concave down on

Range: [0,1]

(k π , (k + 1)π ) where k is any integer.

Even function since

No inflection points

h () −= t cos ( ) 2 −= t cos 2 t = ht ()

so the function is symmetric with respect to the y-axis.

y-intercept: y = 1 ; t-intercepts: x =+ k π

2 where k is any integer. k π

ht '( ) =− 2 cos sin t t ; '( ) ht = at 0 t =

2 k Critical points: π t =

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cos t

function is increasing on the intervals

cos 2 t + 3sin 2 [ t ]

k π + ( / 2), ( π k + 1) π and decreasing on the

intervals [ k ππ , k + ( / 2) π ] .

cos 4 t

+ 2 1 2 sin t

Global maxima hk () π= 1 = 2 4 > 0

cos ⎛ t π ⎞

Global minima h + k π

⎜ over the entire domain. Thus the function is ⎟ = 0

2 2 concave up on ⎜ k π − , k π + ; no inflection ht ''( ) = 2 sin t − 2 cos t =− 2(cos 2 ) t ⎝ ⎟ 2 2 ⎠

ht ''( ) < on 0 ⎜ k π − , k π + ⎟ so h is concave ⎝

points.

=+ t π k π are

No horizontal asymptotes;

down, and ''( ) ht > on 0 ⎜ k π + , k π +

⎝ vertical asymptotes. ⎠

so h

is concave up. ⎛ Inflection points: k ππ 1

No vertical asymptotes; no horizontal asymptotes.

27. Domain: ≈ (– ∞ , 0.44) ∪ (0.44, ∞ ); range: (– ∞ , ∞ )

Neither an even nor an odd function

π t =+ k π

26. Domain: all reals except

2 y -intercept: 0; x-intercepts: 0, ≈ 0.24 Range: [0, ) ∞

74.6092 3 2 y-intercepts:

x – 58.2013 x + 7.82109 x

2 ; is any integer.

; t-intercepts: t

y = 0 = k π where k

fx ′ () =

(7.126 – 3.141) x Even function since

2 2 2 ′ () = 0 when x = 0, ≈ 0.17, ≈ 0.61

g −= t tan ( ) −=− t ( tan ) t = tan t fx

so the function is symmetric with respect to the y-axis.

Critical points: 0, ≈ 0.17, ≈ 0.61

3 ; '( ) gt = when 0 fx ′ () > 0 cos when 0 < x < 0.17 or 0.61 < x t t = k π .

gt '( ) = 2 sec t tan t =

2 2sin t

f (x) is increasing on ≈ [0, 0.17] ∪ [0.61, ∞ ) Critical points: k π

gt () is increasing on k ππ , π ⎞ k

⎢⎣ and decreasing on +⎟ and

(– ∞ , 0] ∪ [0.17, 0.44) ∪ (0.44, 0.61] decreasing on ⎜ k π − , k π ⎥ .

Local minima f(0) = 0, f(0.61) ≈ 0.60; local Global minima ( gk π = ; no local maxima ) 0

maximum f(0.17) ≈ 0.01

′′ 531.665 x 3 – 703.043 x 2 + 309.887 – 24.566 x

f () x =

; (7.126 – 3.141) x 3

f ′′ () x > 0 when x < 0.10 or x > 0.44

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f (x) is concave up on (– ∞ , 0.10) ∪ (0.44, ∞ )

and concave down on (0.10, 0.44); inflection point ≈ (0.10, 0.003)

5.235 x 3 − 1.245 x 2 5.235 x 2 − 1.245 x

so f(x) does not have a horizontal asymptote.

As

– 0.44 , 5.235 3 – 1.245 x 2 → x x → 0.20 while

7.126 – 3.141 x → 0, – so lim fx () =∞ –;

x → 0.44 –

as x →

+ 0.44 , 5.235 3 x – 1.245 2 x → 0.20 while

7.126 – 3.141 x → 0, + so lim fx () =∞ ;

x → 0.44 +

0.44 is a vertical asymptote of f(x).

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40. Let () = 2 fx ax + bx + c , then fx ′ () = 2 ax + b and f ′′ () x = 2. a An inflection point occurs where f ′′ () x changes from positive to negative, but 2a is either always positive or always

negative, so f(x) does not have any inflection points.

( f ′′ () x = 0 only when a = 0, but then f(x) is not a quadratic curve.)

36. y ′ = 5( – 1) ; 20( – 1) ; x 4 y ′′ = x 3 yx ′′ () > 0 41. Let () = 3 fx 2 ax + bx + cx + d , then when x > 1; inflection point (1, 3)

fx ′ () = 3 ax 2 + 2 bx + c and f ′′ () x = 6 ax + 2. b As At x = 1, y′ = 0, so the linear approximation is a long as a ≠ 0 , f ′′ () x will be positive on one horizontal line.

side of x =

and negative on the other side.

is the only inflection point.

42. Let () = 4 fx 3 ax + bx + 2 cx + dx + c , then fx ′ () = 4 ax 3 + 3 bx 2 + 2 cx + d and

f ′′ () x = 12 ax 2 + 6 bx + 2 c 2 = 2(6 ax + 3 bx + c )

Inflection points can only occur when f ′′ () x changes sign from positive to negative and

f ′′ () x = 0. f ′′ () x has at most 2 zeros, thus f(x) has at most 2 inflection points.

43. Since the c term is squared, the only difference occurs when c = 0. When c = 0,

y 3 = 2 x 2 x = x which has domain (– ∞ , ∞ )

and range [0, ∞ ). When c ≠ 0, = 2 2 y 2 x x – c has domain (– ∞ , –|c|] ∪ [|c|, ∞ ) and

range [0, ∞ ).

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The only extremum points are ± . For c c = 0 ,

45. fx () =

, then

there is one minimum, for c ≠ 0 there are two.

( cx 2 – 4) 2

+ cx 2

No maxima, independent of c. No inflection

cx 2 (7 2 − cx 2 )

points, independent of c.

22 If c > 0, 7 fx ′ () = 0 when x = 0, ± . 4() + cx

2 If c < 0, fx ′ () = 0 when x = 0.

fx () = (4 + 222

; () fx ′ = 0 when x =±

c Note that f(x) = 1 (a horizontal line) if c = 0. unless c = 0, in which case f(x) = 0 and

fx ′ () > 0 when x <−

and

If c > 0, f(x) is increasing on ⎢ –, ⎥ and

⎣ cc ⎦

0 << x

, so f(x) is increasing on

decreasing on ⎜ –,– ∞ ⎤⎡ ⎥⎢ ∪ , ∞ ⎞ ⎟ , thus, f(x) has

c ⎦⎣ c ⎠

⎜ ⎜ −∞ − , 0, ⎥⎢ ∪ ⎥ and decreasing on

⎛ 2 1 2 c a global minimum at 2 f c

and a global

maximum of f . 2 c 2 c ⎟ ⎛⎞= . Thus, f(x) has local ⎜⎟ ⎣ ⎦⎣ ⎠

and If c < 0, f(x) is increasing on ⎜ –,–, ∞ ⎥⎢ ∪ ∞ ⎞

and decreasing on ⎡ ,– ⎤ . Thus, f(x) has a

local minimum f (0) =

. If c < 0, fx ′ () > 0

when x < 0, so f(x) is increasing on (– ∞ , 0] and ⎛ 2 global minimum at 1 f

and a global

decreasing on [0, ∞ ). Thus, f(x) has a local

2 1 maximum f (0) =

maximum at f ⎛⎞= ⎜⎟ . 16

. Note that f(x) > 0 and has

⎝⎠ c 4 horizontal asymptote y = 0. 2 cxcx 3 ( 22 – 12)

f ′′ () x =

, so f(x) has inflection

(4 + cx 223 )

points at x = 0, ±

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If c < 0 :

46. fx () =

. By the quadratic formula,

⎡ (4 k + 1) π ( 4 k − 1 ) π ⎤

fx () is decreasing on ⎢

x + 4 x += c 0 when x = –2 ± 4–. c Thus f(x)

has vertical asymptote(s) at x = –2 ± 4– c ⎡ ( 4 k − 1 )( π 4 k − 3 ) π fx ⎤

() is increasing on ⎢

when c ≤ 4. ′ () = ; () ′ = 0 ⎣ 2 c 2 c fx ⎦

–2 – 4 x

fx

( 4 k − 1 ) fx () has local minima at x =

π and local when x = –2, unless c = 4 since then x = –2 is a

2 c vertical asymptote.

For c ≠ 4, fx ′ () > 0 when x < –2, so f(x) is

maxima at x =

where k is an integer.

increasing on (– ∞ , –2] and decreasing on [–2, ∞ ) (with the asymptotes excluded). Thus

If c = 0 , fx () = 0 and there are no extrema.

f (x) has a local maximum at f (–2) =

⎡ ( 4 k − 3 )( π 4 k − 1 ) π ⎤

c = 4, fx ′ () = –

2 c ⎥ ⎣ 2 c ⎦ (– ∞ , –2) and decreasing on (–2, ∞ ).

so f(x) is increasing on

3 fx () is decreasing on ⎢

( x + 2)

⎡ ( 4 k − 1 ) π (4 k + 1) π ⎤

fx () is increasing on ⎢

( 4 k − 1 ) fx () has local minima at x =

π and

local maxima at x =

where k is an

2 c integer.

47. fx () =+ c sin cx . Since c is constant for all x and sin cx is continuous

everywhere, the function fx () is continuous everywhere.

f ' () x =⋅ c cos cx −3π −2π −π

3π x

f ' () x = 0 when cx = k + 1 π or x = k + 1 π

( 2c ) c =1 −

where k is an integer.

f ''

() 2 x =−⋅ c sin cx

( ( k + 2 ) c ) =−⋅ c sin ( c ⋅+ ( k 2 ) c ) =−⋅− c () 1

2 f k ''

In general, the graph of f will resemble the graph of

y = sin x . The period will decrease as c increases

and the graph will shift up or down depending on whether c is positive or negative.

If c = 0 , then fx () = 0 .

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fx '( ) = gx '( 4 )4 x 3 fx '( ) > 0 for on (0,1) x ∪∞ (1, ) fx '( ) < 0 for on ( x −∞ − ∪ − , 1) ( 1, 0)

fx '( ) = 0 for x =− 1, 0,1 since ' is continuous. f

Where x 0 is a root of f ''( ) x = 0 (assume that

there is only one root on (0, 1)).

50. Suppose H ′′′ (1) < 0, then H ′′ () x is decreasing in

a neighborhood around x = 1. Thus, H ′′ () x >0

c c =2

to the left of 1 and H ′′ () x < 0 to the right of 1, so

H (x) is concave up to the left of 1 and concave down to the right of 1. Suppose H ′′′ (1) > 0, then

H ′′ () x is increasing in a neighborhood around x = 1. Thus, H ′′ () x < 0 to the left of 1 and

H ′′ () x > 0 to the right of 1, so H(x) is concave up to the right of 1 and concave down to the left of 1. In either case, H(x) has a point of inflection