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extrema concavity

Total questions: 73

Worksheet time: 2hrs 8mins

Name
Class
Date
1.

If f '(x) changes from positive to negative, f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

2.

If f '(x) changes from negative to positive, then f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

3.

If x is a left endpoint and f '(x) is negative on the interval to the right of x, then f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

4.

If x is a right endpoint and f '(x) is positive on the interval to the left of x, then f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

5.

If x is a left endpoint and f '(x) is positive on the interval to the right of x, then f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

6.

If x is a right endpoint and f '(x) is negative on the interval to the left of x, then f(x) has __________ at x.

a)

a maximum

b)

a minimum

c)

no extrema

7.

Given the sign chart for f '(x), f(x) has ...

a)

a local maximum at x = -2.

b)

a local maximum at x = 4.

c)

local maxima at x = -2 and x = 4.

d)

no extrema.

8.

Given the sign chart for f '(x), f(x) has ...

a)

a local maximum at x = -2.

b)

a local minimum at x = -2 and

a local maximum at x = 4.

c)

a local maximum at x = -2 and

a local minimum at x = 4.

d)

no extrema.

9.

Given the sign chart for f '(x), f(x) has ...

a)

a local maximum at

x = -2.

b)

a local maximum at

x = 4.

c)

a local maximum at x = -2 and

a local maximum at x = 4.

d)

no extrema.

10.

Given the graph of f '(x), f(x) has ...

a)

a local min at x = -6

b)

a local min at x = 2

c)

a local min at x = 2 and

a local max at x = -6

d)

local mins at x = -2, 5 and

a local max at x = 3

11.

Given the graph of f '(x), f(x) has...(mark all that apply)

a)

a local max at x = -9

b)

a local min at x = -9

c)

a local max at x = -6

d)

a local min at x = 0

e)

a local max at x = 7

12.

Given the sign chart for f '(x) and table of f(x), f(x) has ...

a)

a local maximum at (-10,5).

b)

an absolute minimum at (-2,0).

c)

a local minimum at (20,7).

d)

all of the above

13.
When f'(x) changes from negative to positive, there is(are) ...
a)
a maximum.
b)
a minimum.
c)
no extrema.
14.
Use the sign chart for f'(x).
There is(are) ...
a)
a local maximum at x = -2.
b)
a local maximum at x = 4.
c)
local maxima at x = -2 and x = 4.
d)
no extrema.
15.
Given the following graph of f', the derivative of f. For what x-value does f have a relative minimum?
a)
x=-1 and x=4
b)
x=-3
c)
x=1
d)
x=3
16.
When f'(x) changes from positive to negative, there is(are) ...
a)
a maximum.
b)
a minimum.
c)
no extrema.
17.

Find the intervals of concavity for

f(x) = x2 + 2x + 1.

a)

concave up: (-∞,∞)

b)

concave down: (-∞,∞)

c)

concave up: (2, ∞)

concave down: (-∞,2)

d)

concave up: (-∞,2)

concave down: (2, ∞)

18.
 If (a,b) is a local minimum, then what will be true about f'(a)?
a)
It's positive
b)
It's negative
c)
It's zero
d)
Cannot be determined
19.

Find the relative minimum values for f(x)=3x4-2x3-9x2

a)

-1

b)

0

c)

1.5

d)

-1 and 1.5

20.
Identify the interval from the derivative graph where the function is concave up.
a)
(-1,1) & (3,4)
b)
(-3,-2)
c)
(-2,-1)
d)
(-3,-1) & (1,3)
21.
The concavity of a function is described by its _______________.
a)
first derivative
b)
second derivative
c)
third derivative
d)
expression
22.

Over what interval(s) is f(x) concave up? (Be careful this is a graph of f'!)

a)

(-∞, -3) ∪ (1, ∞)

b)

(-3, 1)

c)

(-∞,-3) U (1, ∞)

d)

(-5, 0)U(2, ∞)

23.
Given the velocity function.  Over what interval(s) is the particle moving left?
a)
(0, 3) ∪ (8, 12)
b)
(0, 7) ∪ (10, 12)
c)
(0, 6) ∪ (10, 12)
d)
(3, 8)
24.
The acceleration function is the first derivative of...
a)
position
b)
velocity
c)
calculus
d)
particle motion
25.

If f'(x) = 0 what does that imply about the x value?

a)

It is a critical point, it is a possible max, min, or point of inflection.

b)

That the limit does not exist.

26.
Identify the critical points of the following function:
g(x)=2x3-3x2
a)
x=-1,1
b)
x=0,0
c)
x=0,-1
d)
x=0,1
27.
Over what intervals is f(x) decreasing?
a)
(-∞, -1) ∪ (1, ∞)
b)
(-∞, -√3) ∪ (0, √3)
c)
(-1, 1)
d)
(-√3, 0) ∪ (√3, ∞)
28.
Find the critical points of f(x) = 2x4- 4x2 + 1
a)
x= 0
b)
x = -1, 1
c)
x = -1, 0, 1
d)
no critical points
29.
What will be true at an inflection point?  (select the best answer)
a)
f(x)=0
b)
f'(x)=0
c)
f''(x)=0
d)
The function is undefined
30.

Identify the symmetry of the function

a)

Even

b)

Odd

c)

Neither

31.

Identify the symmetry of the function

a)

Odd

b)

Even

c)

Neither

32.

Identify the symmetry

a)

Odd

b)

Even

c)

Neither

33.

Identify the extrema value

a)

Minimum (-5,-9)

b)

Maximum (-5,-9)

c)

N/A

d)

Minimum (-8,0) & (2,0)

34.
Identify the critical points of the following function:
g(x)=2x3-3x2
a)
x=-1,1
b)
x=0,0
c)
x=0,-1
d)
x=0,1
35.
Given a function, f(x), if f'(x)<0 over a certain interval, then f(x) is ____________ over that interval.
a)
decreasing
b)
increasing
c)
concave up
d)
concave down
36.
Given a function, f(x), if f'(x)>0 over a certain interval, then f(x) is __________ over that interval.
a)
increasing
b)
decreasing
c)
concave up
d)
concave down
37.
Given a function g(x), if g''(x)=0 for a certain value of x, then g(x) has _________ at x.
a)
an inflection point
b)
a maximum
c)
a minimum
d)
a critical point
38.
Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has _____________ at x.
a)
an inflection point
b)
a critical point
c)
a minimum
d)
a maximum
39.
For a function f(x), f''(4)=0 indicates that x=4 is _____________.
a)
an inflection point
b)
a critical point
c)
a relative maximum
d)
a relative minimum
40.
The slope of a function is described by its ____________.
a)
first derivative
b)
second derivative
c)
third derivative
d)
expression
41.
f(x) is pictured. Over what interval is f(x) apparently decreasing?
a)
(infinity, 0)
b)
(-infinity, 0)
c)
(-infinity, infinity)
d)
(0, infinity)
42.
f(x) is pictured. Over which interval is f(x) apparently concave up?
a)
(0, infinity)
b)
(-infinity, infinity)
c)
(-infinity, 0)
d)
nowhere
43.
f''(x) is pictured. Which x values are inflection points of f(x)?
a)
x=-5 and -1
b)
x=-3
c)
x=4
d)
no inflection points
44.
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
45.
Over what intervals is f(x) decreasing?
a)
(-∞, -1) ∪ (1, ∞)
b)
(-∞, -√3) ∪ (0, √3)
c)
(-1, 1)
d)
(-√3, 0) ∪ (√3, ∞)
46.
Find the critical points of f(x) = 2x4- 4x2 + 1
a)
x= 0
b)
x = -1, 1
c)
x = -1, 0, 1
d)
no critical points
47.

If the first derivative of a function changes from positive to negative at a certain point, then that point is a known as a relative minimum.

a)

True (Facts)

b)

False (Not Facts)

48.
f' is given, which could be f?
a)
A
b)
B
c)
C
49.
Find the critical points of f(x) = 2x4- 4x2 + 1
a)
x= 0
b)
x = -1, 1
c)
x = -1, 0, 1
d)
no critical points
50.
What are the intervals of the graph increasing for f(x) = 2x4- 4x2 + 1
a)
(-1,0)
b)
(0,1)
c)
(-∞,-1) and (1,∞)
d)
(0,1) and (-1,0)
51.
a)
It's positive
b)
It's negative
c)
It's zero
d)
Cannot be determined
52.
What is the maximum VALUE of
f(x) = x3 - 3x2 - 1 on the interval [-3, 2]?
a)
0
b)
-1
c)
2
d)
5
53.
a)
A
b)
B
c)
C
54.
If a function has a derivative that is negative, what does that tell you?
a)
The function is increasing
b)
The function is decreasing
55.
a)
A
b)
B
c)
C
d)
D
56.
Derivative means the same thing as
a)
slope of the tangent line
b)
slope of the normal line
c)
exponent
d)
potato
57.
The derivative of a function is its
a)
banana
b)
Maximum/Minimum
c)
Instantaneous rate of change
d)
Common Denominator
58.
Where is the graph continuous yet NOT differentiable?
a)
x = a, b, c, d
b)
x = b, c, d
c)
x = a, b, 
d)
x = b, d
59.
a)
[-2,1]
b)
[-2,3]
c)
[3,5]
d)
[0,1.5] and [3,5]
60.
a)
A
b)
B
c)
C
d)
D
61.
Given a graph of f'', which could be a Point of Inflection?
a)
A
b)
B
c)
C
d)
D
62.
If f''(x) is negative, then f(x) is 
a)
decreasing
b)
increasing
c)
concave down
d)
relative minimum
63.
If f"(x) is positive, then f(x) is 
a)
increasing
b)
positive
c)
relative maximum
d)
concave up
64.
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
65.
If f is continuous and f"(x) is zero or undefined at x=a, then a is always a
a)
critical pt
b)
max
c)
concave up
d)
possible inflection pt
66.
If f"(x) is positive then f'(x) is
a)
increasing
b)
decreasing
c)
concave up
d)
concave down
67.
If f''(x) is negative then f'(x) is
a)
concave down
b)
decreasing
c)
increasing
d)
concave up
68.
If f(x) is increasing, then f'(x) is 
a)
increasing
b)
decreasing
c)
positive
d)
negative
69.
If f(x) is decreasing, then f'(x) is 
a)
positive
b)
concave down
c)
negative
d)
decreasing
70.
If the graph of f' is above the x axis, then f(x) is
a)
increasing
b)
decreasing
c)
positive
d)
concave up
71.
If the graph of f'(x) is below the x-axis, then f(x) is 
a)
decreasing
b)
negative
c)
concave down
d)
concave up
72.
If the graph of f'(x) is below the x-axis for x<a (the left of a) and above the x-axis x>a (the right of a) then f(x) has a
a)
point of inflection
b)
relative min
c)
relative max
d)
layout
73.
If the graph of f'(x) is above the x-axis for x<a (the left of a) and below the x-axis x>a (the right of a) then f(x) has a
a)
relative max
b)
relative min
c)
POI
d)
layout

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