©u u210R143j hKEuHt4as nSwo1f2tVwlagr7eE ELyLVCE.B S EAjl4lk FruiogGhUt5sf srGeisieZrhvPeZdp.9 r 6MtandseI twki0trhY 8I8nbfRiInDiotweS ACSaUlpcnuHlzuHsB.K Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____Date________________Assignment

For each problem, find all points of relative minima and maxima.

1)

y =

x3 − 5

x2 + 7

x − 5

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Relative minimum: (

7

3,

−86

27 )Relative maximum: (1, −2)

For each problem, find all points of relative minima and maxima. You may use the provided graph to

sketch the function.

2)

y =

x3 − 6

x2 + 9

x + 1

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Relative minimum: (3, 1)Relative maximum: (1, 5)

-1-

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For each problem, find all points of relative minima and maxima.

3)

y =

−

x3 − 3

x2 − 1

Relative minimum: (−2, −5)Relative maximum: (0, −1)

4)

y =

x4 − 2

x2 + 3

Relative minima: (−1, 2), (1, 2)Relative maximum: (0, 3)

5)

y =

x4 −

x2

Relative minima: (

−

2

2,

−1

4), (

2

2,

−1

4)

Relative maximum: (0, 0)

6)

y =

−

2

x2 − 4

Relative minimum: (0,

1

2 ) No relative maxima.

7)

y =

(

2

x − 8)

2

3

Relative minimum: (4, 0) No relative maxima.

8)

y =

−

1

5

(

x − 4)

5

3 −

2

(

x − 4)

2

3

Relative minimum: (0,

−

123

2

5)

Relative maximum: (4, 0)

Critical thinking questions:

9) Give an example function

f (

x) where

f

''(0) = 0 and there is no relative minimum or maximum at

x = 0.

Many answers. Ex:

f (

x) = 0,

x,

x3, etc

10) Give an example function

f (

x) where

f

''(0) = 0 and there is a relative maximum at

x = 0.

Many answers. Ex:

f (

x) =

−

x4

-2-

Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com

©y F2G0B1g3K jKAuDtpay SSeoGfGtaw5aMrpe4 3LzLLCV.0 W JA4lXl8 DrviIg3httUsW grleusfe3rSvVeydT.r e uM2ahd8e4 iwVi6tjh6 AIWnOfCiUnUihtNee MC3aKlGcQuXlgufsN.W Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____Date________________Absolute Extrema

For each problem, find all points of absolute minima and maxima on the given closed interval.

1) y = −x3

− 6x2

− 9x + 3; [−3, −1]

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Absolute minimum: (−3, 3)Absolute maximum: (−1, 7)

2) y =8

x2 + 4

; [0, 5]

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Absolute minimum: (5, 8

29)Absolute maximum: (0, 2)

3) y = x3

+ 6x2

+ 9x + 3; [−4, 0]

Absolute minima: (−4, −1), (−1, −1)Absolute maxima: (0, 3), (−3, 3)

4) y = x4

− 3x2

+ 4; [−1, 1]

Absolute minima: (−1, 2), (1, 2)Absolute maximum: (0, 4)

5) y =x

2

3x − 6; [3, 6]

Absolute minimum: (4, 8

3)Absolute maxima: (3, 3), (6, 3)

6) y = (x + 2)2

3; [−4, −2]

Absolute minimum: (−2, 0)Absolute maximum: (−4,

3

4)

-1-

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For each problem, find all points of absolute minima and maxima on the given interval.

7) y = x3

− 3x2

− 3; (0, 3)

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Absolute minimum: (2, −7)No absolute maxima.

8) y = (5x + 25)1

3; [−2, 2]

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

Absolute minimum: (−2, 3

15)Absolute maximum: (2,

3

35)

9) y = x3

− 3x2

+ 6; [0, ∞)

Absolute minimum: (2, 2)No absolute maxima.

10) y = x4 − 2x

2 − 3; (0, ∞)

Absolute minimum: (1, −4)No absolute maxima.

11) y =4

x2 + 2

; (−5, −2]

No absolute minima.

Absolute maximum: (−2, 2

3 )

12) y = −1

6(x + 1)

7

3 +

14

3(x + 1)

1

3; (−5, 0)

Absolute minimum: (−3, −43

2)No absolute maxima.

-2-

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Calculus Maximus WS 5.1: Extrema on an Interval

Page 1 of 8

Name_________________________________________ Date________________________ Period______

Worksheet 5.1—Extrema on an Interval

Show all work. No calculator unless otherwise stated.

1. Let f be the functioned defined on [ ]1,2− by ( ) 2/33 2f x x x= − .

(a) What is its maximum value? (b) What is its minimum value?

2. Sketch the graph of a function f that is continuous on [ ]1,5 and has an absolute minimum at 1x = , an

absolute maximum at 5x = , a local maximum at 2x = , and a local minimum at 4x = .

3. Sketch the graph of f by hand and use your sketch to find the absolute and local extrema of f.

(a) ( )1 , 0 2

2 4, 2 3

x xf x

x x

− ≤ <#= $

− ≤ ≤%(b) ( )

2

2

, -1 0

2 , 0 1

x xf x

x x

! ≤ <#= $

− ≤ ≤#&

Calculus Maximus WS 5.1: Extrema on an Interval

Page 2 of 8

4. Find the critical values of the function

(a) ( ) 4 3 23 4 6x t t t t= + − (b) ( )

2

1

1

zf z

z z

+=

+ +(c) ( ) 2/3 5/3

5g t t t= +

(d) ( ) ( )1g t t t= − (e) ( ) 4 tang θ θ θ= − (f) ( ) lnf x x x=

(g) ( ) 3 2G x x x= − (h) ( ) 2xf x xe= (i) ( ) 2 3g x x= +

Calculus Maximus WS 5.1: Extrema on an Interval

Page 3 of 8

5. Find the absolute extrema of f on the given interval.

(a) ( ) 3 22 3 12 1f x x x x= − − + , [ ]2,3− (b) ( ) ( )

321f x x= − , [ ]1,2− (c) ( )

2

2

4

4

xf x

x

−=

+, [ ]4,4−

(d) ( ) ( )38f t t t= − , [ ]0,8 (e) ( ) sin cosf x x x= + , 0,

3

π" #$ %& '

(f) ( ) 2cosf x x x= − , [ ],π π−

(g) ( )ln x

f xx

= , [ ]1,3 (h) ( ) xf x xe−= , [ ]0,2 (i) ( ) 2x xf x e e− −= − , [ ]0,1

(Hint: factor out 2xe− from ( )f x! )

Calculus Maximus WS 5.1: Extrema on an Interval

Page 4 of 8

6. Show that 5 is a critical value of the function ( ) ( )3

2 5g x x= + − , but g does not have a local extreme

value at 5.

7. Prove that the function ( ) 101 511f x x x x= + + + has neither a local maximum nor a local minimum by

analyzing the derivative function.

MULTIPLE CHOICE

8. Find all the critical values, 0x , of the function ( ) 5 sin5g x x x= + in ( )0,∞ , where 0, 1, 2, n = K .

(A) 0

3 1

5

nx π

+= (B)

05

nx π= (C)

0

2 1

5

nx π

+= (D)

0

4 1

5

nx π

+= (E)

0

1

5

nx π

+=

Calculus Maximus WS 5.1: Extrema on an Interval

Page 5 of 8

9. Find all the critical values of the function ( ) 2cosf x x x= + on the interval ( ),π π− . Hint: think of

where y x= has a critical value, then look at two different cases for ( )f x! .

(A) 5 5

, , 0, , 6 6 6 6

π π π π− − (B) 0x = (C) , 0,

6 6

π π− (D)

2,

3 3

π π−

(E) 5 5

, 0, 6 6

π π− (F)

2,

3 3

π π−

10. Determine the absolute maximum value of ( )2

5 2

14

xf x

x

+=

+ on the interval [ ]2,4− .

(A) 1

18 (B)

13

30 (C)

8

7 (D)

1

2 (E) None

11. Find all the critical values of f when ( ) ( )24/5

5f x x x= − .

(A) 5

0, 7

(B) 10

, 57

(C) 5

, 57

(D) 5

0, , 57

(E) 10

0, 7

(F) 10

0, , 57

Calculus Maximus WS 5.1: Extrema on an Interval

Page 6 of 8

12. Let f be the function defined by ( ) 21 2f x x x= − + on [ ]1,1− .

(i) Find the derivative of f

(A) ( )2

2

1 2

1

xf x

x

−" =

−

(B) ( )2

2

1

2

xf x

x

−" = (C) ( )

2

2

2

1

xf x

x

−" =

−

(D) ( ) 21f x x! = −

(E) ( )2

2

1

xf x

x

! =−

(F) ( ) 22 1f x x x! = −

(ii) Find all the critical points of f in ( )1,1− .

(A) 1

4 (B)

1

4± (C)

1

2± (D)

1

2

± (E) 1

2 (F)

1

2

(iii) Determine the absolute maximum value of f on [ ]1,1− .

(A) 7

2 (B)

5

2 (C) 1 (D)

3

2 (E) 2 (F) 3

Calculus Maximus WS 5.1: Extrema on an Interval

Page 7 of 8

13. Let f be the function defined by ( ) 2sin cosf x x x= − on [ ]0,2π .

(i) Find the derivative of f .

(A) ( ) ( )sin 2cos 1f x x x! = + (B) ( ) ( )cos 2sin 1f x x x! = + (C) ( ) ( )sin 2cos 1f x x x! = − +

(D) ( ) ( )cos 1 2sinf x x x! = − (E) ( ) ( )cos 1 2sinf x x x! = − + (F) ( ) ( )sin 2cos 1f x x x! = −

(ii) Find al the critical values of f in ( )0,2π .

(A) 5 3

, , , 6 2 6 2

π π π π (B)

5, ,

3 3

π ππ (C)

7 3 11, , ,

2 6 2 6

π π π π (D)

2 4 3, , ,

2 3 3 2

π π π π

(E) 2 4

, , 3 3

π ππ (F)

11, ,

6 6

π ππ

(iii) Determine the absolute maximum value of f on [ ]0,2π .

(A) 1− (B) 5

4− (C)

5

4 (D) 1 (E)

3

4− (F)

3

4

Calculus Maximus WS 5.1: Extrema on an Interval

Page 8 of 8

14. Let f be the function defined by ( )2

12

16

f x xx

! "= −$ %

& ', 0x ≠ . Determine the absolute maximum value

of f on ( ], 1−∞ − . Hint: find the domain of f, the critical values of f, then look at the sign of f ! for all

values in the specified interval.

(A) 17

8− (B) No max value (C)

3

2− (D)

3

2 (E)

17

8

15. An advertisement is run to stimulate the sale of cars. After t days, 1 48t≤ ≤ , the number of cars sold is

given by ( ) 2 34000 45N t t t= + − . On what day does the maximum rate of growth (that’s ( )N t! ) of

sales occur?

(A) day 17 (B) day 13 (C) day 15 (D) day 16 (E) day 14