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Connecting f, f', and f''

Total questions: 64

Worksheet time: 11hrs 40mins

Name
Class
Date
1.

f''(x) is pictured. Which x values are inflection points of f(x)?

a)

x=-5 and -1

b)

x=-3

c)

x=0

d)

no inflection points

2.

For a function g(x), g'(-2)=0 indicates that x=-2 is ________________.

a)

an inflection point

b)

a critical point

c)

a local maximum

d)

a local minimum

3.
For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.
a)
increasing
b)
decreasing
c)
concave up
d)
concave down
4.

Which of the following describes an interval of f(x) that is both decreasing and concave up?

a)

f '(x) < 0 and f "(x) < 0

b)

f '(x) < 0 and f "(x) > 0

c)

f '(x) > 0 and f "(x) > 0

d)

f '(x) > 0 and f "(x) < 0

5.

For a function g(x), g''(3) = -8 indicates that g(x) is ____________ at x=3.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

6.

f(x) is pictured. Inflection points are most likely at which x values?

a)

x=0 and 1.5

b)

x=2 and 2.5

c)

x=2.25

d)

x=1

7.

When f'(x) changes from positive to negative, there is ...

a)

a local maximum.

b)

a local minimum.

c)

no maximum or minimum

d)

a point of inflection

8.

When f'(x) changes from positive to negative, there is ...

a)

a local maximum.

b)

a local minimum.

c)

no maximum or minimum

d)

a point of inflection

9.

What will be true at an inflection point? (select the best answer)

a)

f(x)=0 and there is a sign change

b)

f'(x)=0 and there is a sign change

c)

f''(x)=0 and there is a sign change

d)

The function is increasing then decreasing

10.

The graph of f', the derivative of f is shown to the left. Determine all the intervals on which the function f is concave down on the interval (-9,9).

a)

(−4,0)∪(2,9)\left(-4,0\right)\cup\left(2,9\right)

b)

(−5,−1)∪(1,4)\left(-5,-1\right)\cup\left(1,4\right)

c)

(−9,−4)∪(0,2)\left(-9,-4\right)\cup\left(0,2\right)

d)

(−9,−5)∪(−1,1)∪(4,9)\left(-9,-5\right)\cup\left(-1,1\right)\cup\left(4,9\right)

11.

From the graph on the left it can be seen that the function f on the interval (-2,0) is positive, increasing and concave down. Based on these observations it can be concluded that:

a)

f' is positive and increasing, f" is negative

b)

f' is positive and decreasing, f" is negative

c)

f' is negative and decreasing, f" is negative

d)

f' is positive and decreasing, f" is positive

12.

From the graph on the left it can be seen that the graph f" on the interval (-9,-7) is positive, decreasing and concave up. Based on these observations it can be concluded that:

a)

f is concave up, f' is increasing

b)

f is increasing, f' is positive

c)

f is concave down, f' is decreasing

d)

f is concave up, f' is negative

13.

Which of the following could be the graph of f ' , the derivative of f ?

a)
b)
c)
d)
e)
14.

f(x) has a point of inflection, therefore f''(x) has _?

a)

relative max

b)

relative min

c)

equals zero or undefined

d)

Point of Inflection

15.

f'(x) has a min, therefore f''(x) has _?

a)

relative max

b)

relative min

c)

equals zero or undefined

d)

Point of Inflection

16.

f'(x) changes from concave up to concave down, therefore f''(x) has _?

a)

relative max

b)

relative min

c)

equals zero or undefined

d)

Point of Inflection

17.

f'(x) changes from concave up to concave down, therefore f''(x) has _?

a)

relative max

b)

relative min

c)

equals zero or undefined

d)

Point of Inflection

18.
Over what interval(s) is f(x) decreasing?
a)
(-3, 1)
b)
(-∞, -5) ∪ (0, 2)
c)
(-∞, -3) ∪ (1, ∞)
d)
(-5, 0) ∪ (2, ∞)
19.
Given this graph of f'(x), on what interval(s) is f increasing?
a)
[-1,2]
b)
[0,1] U [3,4]
c)
[-1,0] U [1,3]
d)
[2,4]
20.
If f"(x) is positive then f'(x) is
a)
increasing
b)
decreasing
c)
concave up
d)
concave down
21.
If f(x) is continuous and f'(x) is zero or undefined at x=a, then a is ALWAYS called a
a)
critical point
b)
absolute max
c)
absolute min
d)
inflection point
22.
a)
[-2,1]
b)
[-2,3]
c)
[3,5]
d)
[0,1.5] and [3,5]
23.
Where does f(x) have a point of inflection?
a)
x = -3
b)
x = 3,-3
c)
x= -1
d)
x = -1,3
24.
Given a graph of f'', which could be a Point of Inflection?
a)
A
b)
B
c)
C
d)
D
25.

What is true about the following? (This is an f graph)

a)

A

b)

B

c)

C

d)

D

26.

What is true about the following? (This is an f graph)

a)

A

b)

B

c)

C

d)

D

27.

What is true about the following? (This is an f graph)

a)

A

b)

B

c)

C

d)

D

28.

Where f'' is positive, f _____________

a)

is concave up

b)

is increasing

c)

is decreasing

d)

is concave down

29.

Which emoji best describes how you are feeling today?

a)

😁

b)

😊

c)

😢

d)

😟

e)

😤

30.

Where f'' is positive, f _____________

a)

is concave up

b)

is increasing

c)

is decreasing

d)

is concave down

31.

Where f has a local minimum, f'' ...

a)

is positive

b)

is increasing

c)

is concave up

d)

changes signs

32.

Where f is concave up, f' ...

a)

is positive

b)

is concave up

c)

is increasing

d)

has a local minimum

33.

Where f has a point of inflection, f''...

a)

changes signs

b)

equals zero or does not exist

c)

has a local minimum

d)

has a local maximum

34.

Where f' changes from negative to positive, f...

a)

is concave down

b)

has a local maximum

c)

has a local minimum

d)

is concave up

35.

Where f'' is negative, f...

a)

is decreasing

b)

is concave down

c)

has a local maximum

d)

has a local minimum

36.

Where f' is concave down, f'' ...

a)

has a local maximum

b)

equals 0

c)

is negative

d)

is decreasing

37.

Where f' is increasing, f'' ...

a)

is positive

b)

is concave up

c)

is negative

d)

is concave down

38.

Where f' is positive, f...

a)

is increasing

b)

is concave up

c)

has a local maximum

d)

is negative

39.

Where f has a local minimum, f'...

a)

changes from negative to positive

b)

is increasing

c)

changes from positive to negative

d)

is decreasing

40.

Where f has a point of inflection, f'...

a)

has a relative max/min

b)

changes concavity

c)

changes from increasing to decreasing or vice-versa

d)

equals 0 or does not exist

41.

Where f' is negative, f...

a)

is decreasing

b)

is increasing

c)

is concave up

d)

is concave down

42.

Where f'' changes signs, f...

a)

has a point of inflection

b)

changes concavity

c)

has a local max/min

d)

equals 0 or does not exist

43.

Where f'' is positive, f'...

a)

is increasing

b)

is concave up

c)

has a local maximum

d)

has a local minimum

44.

Where f' changes from positive to negative, f...

a)

has a local maximum

b)

is concave down

c)

has a local minimum

d)

is concave up

45.

Where f'' is negative, f'...

a)

is decreasing

b)

is increasing

c)

is positive

d)

is negative

46.

Where f'' is increasing, f'...

a)

is concave up

b)

is concave down

c)

is positive

d)

is negative

47.

Where f is decreasing, f'...

a)

is negative

b)

is positive

c)

is concave up

d)

is concave down

48.

Where f' is increasing, f...

a)

is concave up

b)

is concave down

c)

is positive

d)

is negative

49.

Where f has a local maximum, f'...

a)

changes from negative to positive

b)

is decreasing

c)

equals zero

d)

changes from positive to negative

50.

Where f' has a point of inflection, f''...

a)

has a local max/min

b)

equals zero or does not exist

c)

changes concavity

d)

changes signs

51.

Where f' is concave up, f''...

a)

is increasing

b)

is decreasing

c)

is positive

d)

is negative

52.

Where f' is decreasing, f...

a)

is concave down

b)

is concave up

c)

is positive

d)

is negative

53.

Where f is increasing, f'...

a)

is positive

b)

is negative

c)

is concave up

d)

is concave down

54.

Where f'' changes from positive to negative, f' ...

a)

has a local maximum

b)

is concave down

c)

has a local minimum

d)

is concave up

55.

Where f' has a local extremum, f...

a)

has a point of inflection

b)

changes signs

c)

equals zero or does not exist

d)

is decreasing

56.

Where f'' is decreasing, f'...

a)

is concave down

b)

is concave up

c)

is positive

d)

is negative

57.

Where f has a local maximum, f''...

a)

is negative

b)

is positive

c)

changes from positive to negative

d)

changes from negative to positive

58.

Where f is concave down, f' ...

a)

is decreasing

b)

is increasing

c)

is positive

d)

is negative

59.

Where f' has a local maximum, f''...

a)

changes from positive to negative

b)

is decreasing

c)

changes from negative to positive

d)

is increasing

60.

Where f is concave up, f''...

a)

is positive

b)

is negative

c)

is increasing

d)

is decreasing

61.

Where f' is decreasing, f''...

a)

is negative

b)

is positive

c)

is concave up

d)

is concave down

62.

Where f'' has a local extremum, f'...

a)

has a point of inflection

b)

changes signs

c)

equals zero or does not exist

d)

is increasing

63.

Where f is concave down, f''...

a)

is negative

b)

is positive

c)

is increasing

d)

is decreasing

64.

Where f'' changes from negative to positive, f'...

a)

has a local minimum

b)

is concave up

c)

has a local maximum

d)

is concave down

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