#### Question Details

I need help with all the problems attached below (all 25 problems).

problem 1-25

hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

This print-out should have 25 questions.

Multiple-choice questions may continue on

the next column or page ? find all choices

before answering.

001 1 5. c = 6 , 7

6. c = 12 4.0 points 003 4.0 points The function f is defined on [1, 3] by

Determine if Rolle?s Theorem can be applied to

f (x) = x2 + 3x ? 18

x+3 on the interval [?6, 3], and if it can, find all

numbers c satisfying the conclusion of that

theorem. 2 At which point c in (1, 3), if any, does

f (3) ? f (1)

= f ? (c)

2

hold?

1. c = 1. c = ?1

2. c = ?3, ? 13

5 2. no such c exists 3

2 3. c = 2 3. c = ?3, ?15

4. Rolle?s Theorem not applicable

3

5. c = ?

2 4. c = 7

3 5. c = 5

2

004 6. c = ?3

002 1 f (x) = 2x + (x ? 1) 3 (3 ? x) 3 . 4.0 points Determine if the function

?

f (x) = x 18 ? x

satisfies the hypotheses of Rolle?s Theorem

on the interval [0, 18], and if it does, find all

numbers c satisfying the conclusion of that

theorem. Find the value of f (2) when f, g are differentiable functions such that

?

f

f (x)g ?(x)

= ?

,

g(x) > 0 ,

g

(g(x))2

for all x, while f (0) = 6 .

1. None of these

2. f (2) = 12 1. c = 7

2. hypotheses not satisfied 4.0 points 3. f (2) = 6 3. c = 12 , 13 4. f (2) = 4 4. c = 13 5. f (2) = 2 hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

005 4.0 points Which of the following functions

A. f (x) = 1

on [0, 2],

x?1 B. f (x) = x1/3 on [0, 1],

C. f (x) = |x| on [0, 1],

satisfy the hypotheses of the MVT? The derivative of a function f is given for

all x by

f ? (x) = (2x2 + 6x ? 8) 1 + g(x)2

where g is some unspecified function. At

which point(s) will f have a local maximum?

1. local maximum at x = 4

2. local maximum at x = ?4

3. local maximum at x = ?4, 1

4. local maximum at x = ?1 1. A only 5. local maximum at x = 1 2. B and C only 008 3. C only 4.0 points Let f be the function defined by

4. all of them f (x) = 1 + x2/3 . 5. none of them Consider the following properties: 6. A and C only A. concave up on (??, 0) ? (0, ?) ; 7. B only B. has local minimum at x = 0 ; 8. A and B only C. derivative exists for all x 6= 0 . 006 Which does f have?

4.0 points 1. A and C only How many real roots does the equation

x5 + 3x + 5 = 0 2. B and C only

3. C only have?

1. exactly four real roots

2. no real roots 4. A and B only

5. A only 3. exactly three real roots

4. exactly one real root

5. exactly two real roots

007 4.0 points 2 6. B only

7. None of them

8. All of them hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

009 3

1 4.0 points 1. The derivative, f ? , of f has graph

f ? (x) a b c 2. f ? (x)

graph of f ?

1

Use it to locate the critical point(s) x0 at

which f has a local maximum?

1. 3. x0 = c f ? (x) 2. x0 = a

3. x0 = b , c 4. x0 = c , a

1 5. x0 = b

6. x0 = a , b , c 7. x0 = a , b 8. 1

4.

f ? (x) none of a , b , c

010 4.0 points If f is decreasing and its graph is concave

up on (0, 1), which of the following could be

the graph of the derivative, f ? , of f ? 011 4.0 points The graph of a twice-differentiable function

f is shown in hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

C. f is concave up on (0, 2).

1. A only

2. A and C only b 1 3. none of them

4. A and B only

5. C only Which one of the following sets of inequalities

is satisfied by f and its derivatives at x = 1?

1. f ? (1) < f ?? (1) < f (1)

?? 6. all of them

7. B only ? 2. f (1) < f (1) < f (1)

8. B and C only 3. f ? (1) < f (1) < f ?? (1)

4. f ?? (1) < f (1) < f ? (1) 013 5. f (1) < f ? (1) < f ?? (1) On which interval(s) is 6. f (1) < f ?? (1) < f ? (1)

012 4.0 points f (x) = x4 ? 2x2 ? 8 4.0 points decreasing? When Sue uses first and second derivatives

to analyze a particular continuous function

y = f (x) she obtains the chart

? ?? y

y

y

x < ?3

+

?

x = ?3

4

0

?3 < x < 0

?

?

x=0

1

?1

0<x<2

?

+

x=2

?1 DNE

x>2

+

+ Which of the following can she conclude from

her chart?

A. f is concave down on (??, 0).

B. f has a point of inflection at x = 0. 1. (?? , 0 )

2. ( 0 , ? )

3. ( ?? , ?1 ) ,

4. (?1 , 0 ) , (1, ?) (1, ?) 5. (?1, 1 )

6. ( ?? , ?1 ) ,

014 (0, 1)

4.0 points When the graph of f is 4 hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

8

7

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8 6

4

2

?6 ?4 ?2

?2 2 4 6 8 10 ?4

?6 which of the following is the graph of f ?? ?

8

7

6

1. 54

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8

8

7

6

2. 54

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8

7

6

3. 54

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8 4 8

7

6

4. 54

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8

8

7

6

5. 54

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8 4

?4 4 8 ?4 4 8 4 8 ?4 4

?4

?4 015

?4 5 4.0 points A polynomial function f has the properties

(i) f (?7) = 0 = f (?4),

(ii) f ?? (x) > 0 on (?7, ?4). Which of the following statements is a consequence of these properties?

1. f (x) ? 0 on [?7, ?4] 4

?4 4 8 2. f ? (x) ? 0 on (?7, ?4)

3. f (x) = 0 on [?7, ?4] ?4 4. f ? (x) ? 0 on (?7, ?4)

5. f (x) ? 0 on [?7, ?4]

4

016

?4 4

?4 8 4.0 points If the graph of

f (x) = ax3 + bx2 + cx + d

has a local maximum at (0, 1) and a local

minimum at (2, ?3), compute the value of

f (1). hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

1. f (1) = 1 019 (part 1 of 2) 4.0 points 2. f (1) = 2 Lete f be the function defined by 3. f (1) = 3 f (x) = 4. f (1) = 0 017 4.0 points Find all values of x at which the graph of

1

f (x) = x3 ? 2x2 + 7

3

has a point of inflection.

4

3 4

3 2. 2x

?1 x2 3. ? (x2 (x2 2x

? 1)2 2x

? 1)2 (x2 (x + 2)

? 1)(x ? 1) 020 (part 2 of 2) 4.0 points 5. x = 2 (ii) Find the interval(s) on which f is increasing. 6. x = 0, ?4

018 1. (??, 0)

4.0 points Find the interval(s) where

4 4

x ? 2x3 ? 2x2 ? x + 8

3

is concave down.

1

1. ?1, ?

4

1

2. ??, ?1 , ? , ?

4

1

3. ? , 1

4

1

, 1, ?

4. ??,

4

1

5. ??, ? , 1, ?

4

f (x) = (x2 5. ? 3. x = ?2 x

? 1)2 1. ? 4. 2. x = 0, 4 4. x = x2

.

x2 ? 1 (i) Which of the following functions is the

derivative of f ? 5. f (1) = ?1 1. x = 0, ? 6 2. (??, ?1), (?1, 0)

3. (?1, 1), (1, ?)

4. (?1, ?)

5. (??, ?1), (?1, 1)

021 4.0 points Let f be the function defined by

f (x) = x ? 2 sin(x), 0 ? x ? ?. On which interval(s) is f increasing? hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

1. 1

0, ?

3 4. x = ? 3. 1

0, ?

3 1

21/2 5. x = ?1 2. (0, ?)

7 ? 2

?, ?

3 024 (part 3 of 4) 4.0 points

Locate all the local maxima of f .

1 5

?, ?

4.

6 6

1

5.

?, ?

3

1. x = ? 1

21/4 2. x = ?1, ? 022 (part 1 of 4) 4.0 points 1

21/4 3. none of these Let f be the function defined by

f (x) = x2 + 1

x2 on (??, 0).

Find the derivative of f

2x4 + 1

1. f ? (x) =

x3

2(x4 + 1)

2. f (x) =

x3 4. x = ? 1

21/2 5. x = ?1

025 (part 4 of 4) 4.0 points

Determine the absolute minimum value

minf of f . ? 3. f ? (x) = 2(x4 ? 1)

x3 1. minf = 4

2. none of these 2x4 ? 1

4. f (x) =

x3 3. minf = 1 4x3 + 1

x4 4. minf = 3 ? 5. f ? (x) = 023 (part 2 of 4) 4.0 points

Find all the critical points of f .

1. x = ?1, ? 1

21/4 2. x = none of these 3. x = ? 1

21/4 5. minf = 2

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DATE ANSWEREDJan 30, 2021

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