Deck 14: Curve Sketching

Question 1

The graph of a function is given.Find
(a)the open intervals on which the function is increasing or decreasing;
(b)the coordinates of all relative extrema.

Answer 1

(a)f(x)is increasing on (-∞,-2)and (2,∞); f(x)is decreasing on (-2,0)and (0,2)
(b)relative extrema at (-2,1)and (2,1)

Question 2

Find all the critical values of f(x)=
.

Answer 2

± 3

Question 3

If f(x)= 2
+ 3
- 36x + 1,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Answer 3

increasing on (-∞,-3)and (2,∞); decreasing on (-3,2)

Question 4

Find all the critical values of f(x)=
- 8
+ 3.

Question 5

Find all the critical values of f(x)=
.

Question 6

The graph of a function is given.Find
(a)the open intervals on which the function is increasing or decreasing;
(b)the coordinates of all relative extrema.

Question 7

Let f(x)=
+
-
.Determine the intervals on which f is
(a)Determine the intervals on which f is increasing.
(b)Determine the intervals on which f is decreasing.
(c)Based on your answers to parts (a)and (b),find the values of x for which f has relative maxima.
(d)Based on your answers to parts (a)and (b),find the values of x for which f has relative minima.

Question 8

The graph of a function is given.Find
(a)the open intervals on which the function is increasing or decreasing;
(b)the coordinates of all relative extrema.

Question 9

If f(x)=
,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 10

The graph of a function is given.Find
(a)the open intervals on which the function is increasing or decreasing;
(b)the coordinates of all relative extrema.

Question 11

If f(x)= 7
+ 3x - 2,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 12

Find all the critical values of f(x)= x ln x.

Question 13

Find all the critical values of y =
-
- x + 2.

Question 14

If f(x)=
-
,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 15

If f(x)=

,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 16

If f(x)=
-
- 15x - 7,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 17

If f(x)=
,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 18

If f(x)= (
,determine the intervals on which f is increasing and the intervals on which f is decreasing.

Question 19

Find all the critical values of f(x)=
.

Question 20

Let f(x)=
.
(a)Determine the intervals on which f is increasing.
(b)Determine the intervals on which f is decreasing.
(c)Based on your answers to parts (a)and (b),find the values of x for which f has relative maxima.
(d)Based on your answers to parts (a)and (b),find the values of x for which f has relative minima.

Question 21

The function y = + 15
- 33x has a relative maximum when x =

A) 0.
B) -1.
C) 1.
D) 11
E) -11.

Question 22

Let f(x)= x(x -
.
(a)Determine the intervals on which f is increasing.
(b)Determine the intervals on which f is decreasing.
(c)Based on your answers to parts (a)and (b),find the values of x for which f has relative maxima.
(d)Based on your answers to parts (a)and (b),find the values of x for which f has relative minima.

Question 23

Use your graphing calculator to graph y = ln(x)+
.Inspect the graph to estimate where the relative extrema occur.Then use the first-derivative test to find the relative extrema.

Question 24

Let f(x)=
.
(a)Determine the intervals on which f is increasing.
(b)Determine the intervals on which f is decreasing.
(c)Based on your answers to parts (a)and (b),find the values of x for which f has relative maxima.
(d)Based on your answers to parts (a)and (b),find the values of x for which f has relative minima.

Question 25

How many critical values does the function f(x)= 3 + 4
Have?

A) none
B) one
C) two
D) three
E) four

Question 26

The function f(x)= - 6x + 8 is decreasing on

A) (3, ∞).
B) (-3, 3).
C) (2, 4).
D) (-∞, 2) and (4, ∞).
E) (-∞, 3).

Question 27

The function f(x)= 4 - 10
- 8x + 3 is decreasing on

A)
B)
C) (2, ∞)
D)
E) (-∞, 2).

Question 28

Let f(x)=
+
.
(a)Determine the intervals on which f is increasing.
(b)Determine the intervals on which f is decreasing.
(c)Based on your answers to parts (a)and (b),find the values of x for which f has relative maxima.
(d)Based on your answers to parts (a)and (b),find the values of x for which f has relative minima.

Question 29

Determine the intervals on which the function is increasing and on which it is decreasing.Also determine the points of relative maxima and relative minima.
y =
+
- 3x + 7

Question 30

The function y = - 8
Has a relative maximum when x =

A) 1.
B) 2.
C) 0.
D) -1.
E) -2.

Question 31

Use your graphing calculator to graph y =
- 6
+ 9x- 2.Inspect the graph to estimate where the relative extrema occur.Then use the first-derivative test to find the relative extrema.

Question 32

The cost equation for a cookie store is given by C(x)=
- 6
+ 250,where x is the number of cookies made (in dozens)and C(x)is the cost in dollars.Use the first-derivative test to find when relative extrema occur.

Question 33

Determine the intervals on which the function is increasing and on which it is decreasing.Also determine the points of relative maxima and relative minima.
f(x)= 16
- 5x

Question 34

The profit equation for a taco stand is given by P(x)= -0.4
+ 100x - 100,where x is the number of tacos sold,and P(x)is the profit in dollars.Use the first-derivative test to find when relative extrema occur.

Question 35

Determine the intervals on which the function is increasing and on which it is decreasing.Also determine the points of relative maxima and relative minima.
f(x)=

Question 36

The cost equation for a bakery is given by C(x)= 10(x -
- 120(x - 2)+ 450,where x is the number of doughnuts made (in dozens),and C(x)is the cost in dollars.Use the first-derivative test to find when relative extrema occur.

Question 37

Find all the critical values of f(x)=
- 2
.

Question 38

Determine the intervals on which the function is increasing and on which it is decreasing.Also determine the points of relative maxima and relative minima.
f(x)= (
- 3x +

Question 39

Find all the critical values of f(x)= 12
- 65
+ 45x + 60.

Question 40

Determine the intervals on which the function is increasing and on which it is decreasing.Also determine the points of relative maxima and relative minima.
f(x)=

Question 41

Given y = f(x)= 2
+12
- 7 on the interval
,find the values of x at which absolute maxima and absolute minima occur.

Question 42

On the interval ,the function y = 4 +
-
Has an absolute maximum when x =

A) 0.
B) 1.
C) -1.
D)
E) -

Question 43

If C = -3x - 2
+
is a cost function,sketch the graph of this function with the aid of intercepts,symmetry,and the first-derivative test.

Question 44

If C = -4x +
is a cost function,sketch the graph of this function with the aid of intercepts,symmetry,and the first-derivative test.

Question 45

Use your graphing calculator to estimate the relative extrema of the function
(estimate to 3 decimal places).

Question 46

A drug is injected into a patient's bloodstream.The concentration of the drug in the bloodstream t hours after the injection is approximated by C(t)=
.Find the relative extrema and use this to determine when the drug is at its greatest concentration.

Question 47

Sketch the graph of y =
- 4x with the aid of intercepts,symmetry,and the first-derivative test.Use your Graphing Calculator to verify your solution.

Question 48

Use your graphing calculator to estimate the relative extrema of the function y =
-
-
(estimate to 3 decimal places).

Question 49

On the interval ,the function y =
-
Has

A) an absolute max. at x = 2 and an absolute min. at x = 1.
B) an absolute min. at x = 1 and no absolute max.
C) an absolute max. at x = 0 and an absolute min. at x = 1.
D) an absolute max. at x= 2 and an absolute min. at x = 0.
E) no absolute max. and no absolute min.

Question 50

Find the absolute extrema for y =
on the interval
and where they occur.

Question 51

A new movie has its monthly revenue given by R(x)=
where x is the number of days after its release and R is in millions of dollars.Find the relative extrema and use this information to determine which day will bring the greatest revenue.

Question 52

Find the absolute extrema for y =
+
- 3x + 7 on the interval
and where they occur.

Question 53

Sketch the graph of y =
- 3
+ x + 1 with the aid of intercepts,symmetry,and the first-derivative test.Use your graphing calculator to verify your solution.

Question 54

A new book has its monthly revenue given by R(x)=
where x is the number of months after its release and R is in thousands of dollars.Find the relative extrema and use this information to determine which month will bring the greatest revenue.

Question 55

Given y = f(x)=
- 3
+ 10 on the interval
,find the values of x at which absolute maxima and absolute minima occur.

Question 56

Sketch the graph of y =
- 7x with the aid of intercepts,symmetry,and the first-derivative test.Use your graphing calculator to verify your solution.

Question 57

On the interval ,the function y =
+ 3
- 9x + 27 has an absolute maximum when x =

A) 2.
B) 0.
C) 1.
D)
E) none of the above

Question 58

Given y = f(x)= 4
- 4x + 5 on the interval
,find the values of x at which absolute maxima and absolute minima occur.

Question 59

If C = 4x - 4
+
is a cost function,sketch the graph of this function with the aid of intercepts,symmetry,and the first-derivative test.

Question 60

If C = 4x - 5
+
is a cost function,sketch the graph of this function with the aid of intercepts,symmetry,and the first-derivative test.

Question 61

Find the intervals where the graph of y = f(x)is concave up and where it is concave down.Do not sketch its graph.Also determine its points of inflection.f(x)= 16
- 5x.

Question 62

Let f(x)= 3
- 10
+ 7x.Determine the intervals on which f is (a)concave up and (b)concave down.
(c)Find the x-values of all inflection points.

Question 63

The function y = +
+ x - 3 has how may inflection points?

A) none
B) one
C) two
D) three
E) four

Question 64

If f(x)= + 2
- 4x + 10,then f is

A) decreasing on (-2, ∞), concave down on , and has a relative maximum when x = -2.
B) decreasing on , concave up on , and has a relative minimum when x = -
C) increasing on , concave down on , and has a relative minimum when x = -2.
D) increasing on , concave down on , and has a relative maximum when x =
E) increasing on , concave up on (-∞, ∞), and has a relative minimum when x = -2.

Question 65

Let f(x)=
+ 5
.Determine the intervals of which f is (a)concave up and (b)concave down.(c)Find the x-values of all inflection points.

Question 66

The function f(x)= is concave up on the interval(s)

A) (0, ∞).
B) (-∞, 0).
C) (1, ∞).
D) (-∞, 0) and (0, 1).
E) (-∞, -1).

Question 67

Let y =
- 4
.
(a)Determine y' and y''.
(b)Determine intervals on which the function is increasing; determine intervals on which the function is decreasing.
(c)Determine the coordinates of all relative maximum and relative minimum points.
(d)Determine intervals on which the function is concave up; determine intervals on which the function is concave down;
(e)Determine the coordinates of all inflection points.
(f)With the aid of the information obtained in parts (a)-(e),give a reasonable sketch of the curve.

Question 68

Use your graphing calculator to find the absolute extrema of f(x)=
- 4
+ 3x + 2 on
(round answers to 3 decimal places).

Question 69

If f(x)= (x + 4),then f has an inflection point when x =

A) 0.
B) -1.
C) -3.
D) -5.
E) -6.

Question 70

If f(x)= - 6
+ 3,then f has an inflection point when x =

A) 0.
B) 1.
C) 2.
D) 3.
E)

Question 71

Let y =
- 3
- 9x + 10.
(a)Determine y' and y''.
(b)Determine intervals on which the function is increasing; determine intervals on which the function is decreasing.
(c)Determine the coordinates of all relative maximum and relative minimum points
(d)Determine intervals on which the function is concave up; determine intervals on which the function is concave down;
(e)Determine the coordinates of all inflection points.
(f)With the aid of the information obtained in parts (a)-(e),give a reasonable sketch of the curve.

Question 72

Let f(x)=
.Find the x values of (a)all relative maxima,(b)all relative minima,and (c)all inflection points.

Question 73

If y = x(x -
,find the x-values of all inflection points.

Question 74

Find the intervals where the graph of y = f(x)is concave up and where it is concave down.Do not sketch its graph.Also determine its points of inflection.f(x)=

Question 75

Let f(x)=
.Determine the intervals of which f is (a)concave up and (b)concave down.(c)Find the x-values of all inflection points.

Question 76

The function f(x)= is concave down on the interval(s)

A) (-∞, -3).
B) (-3, ∞).
C) (-3, 0).
D) (-∞, -3) and (2, ∞).
E) (-∞, ∞).

Question 77

If f(x)= + 3
- 24x + 8,then f is

A) decreasing on (-∞, 4), concave up on (-1, ∞), and has no relative minimum point.
B) decreasing on (1, 2), concave up (0, ∞), and has a relative minimum when x = -4.
C) increasing on (2, ∞), concave up on (-∞, -1), and has a relative minimum when x = 2.
D) increasing on (-4, 2), concave down on (-∞, ∞), and has a relative maximum when x = 2.
E) decreasing on (-4, 2), concave down on (-∞, -1), and has a relative maximum when x = -4.

Question 78

Use your graphing calculator to find the absolute extrema of
on
(round answers to 3 decimal places).

Question 79

If y =
+ 4
- 18
- 3x + 4,find the x-values of all inflection points.

Question 80

If f(x)= - 7
+ 2x - 5,then f is concave down on the interval

A)
B)
C)
D)
E) (-∞, ∞).