Question 1

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 12 x - 5$ .

Accepted Answer

A)  None 
B)  -2, 2 
C)  0, -2, 2 
D)  $3 \sqrt { 5 }$ 
A)  None 
B)  -2, 2 
C)  0, -2, 2 
D)  $3 \sqrt { 5 }$

Question 2

Find two non-negative numbers x and y whose sum is 4 and are such that the xy2 is as large as possible.

Accepted Answer

Question 3

Find the absolute maximum and minimum of the function $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval 0 $\le$ x $\le$ 3.

Accepted Answer

Maximum is

Question 4

Find two non-negative numbers x and y whose sum is 10 and are such that xy2 is as large as possible.

Accepted Answer

A)  $\frac { 10 } { 3 } \text { and } \frac { 20 } { 3 }$ 
B)  10 and 20 
C)  5 and 5 
D)  9 and 1 
A)  $\frac { 10 } { 3 } \text { and } \frac { 20 } { 3 }$ 
B)  10 and 20 
C)  5 and 5 
D)  9 and 1

Question 5

The derivative of function shown below is negative on the interval shown.

Accepted Answer

A) True 
 B)False

Question 6

Name the vertical and horizontal asymptotes of the given graph.

Accepted Answer

Vertical asymptotes:

Question 7

Determine where the second derivative in the function graphed below is positive.

Accepted Answer

The answer of Determine where the second derivative in the...

Question 8

The function $f ( x ) = x ^ { 6 } - 5 x ^ { 3 } + 2$ has a relative maximum at x = 1.

Accepted Answer

A) True 
 B)False

Question 9

Find the absolute maximum of the function $f ( x ) = x ^ { 5 } - x ^ { 4 }$ on the interval -1$\le$ x $\le$  1.

Accepted Answer

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

Question 10

Find the intervals of increase and decrease for $f ( x ) = 4 x ^ { 3 } + 6 x ^ { 2 } - 72 x - 3$ .

Accepted Answer

Increasing on x < -3

Question 11

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

Accepted Answer

A) For x > 2 B) For x -2 D) For x < 2 A) For x > 2 B) For x -2 D) For x < 2

Question 12

Determine where the graph of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ is concave up and concave down.

Accepted Answer

Concave up for x < -

Question 13

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1$\le$ x$\le$ 5.

Accepted Answer

A)  5 
B)  -4 
C)  -1 
D)  0 
A)  5 
B)  -4 
C)  -1 
D)  0

Question 14

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 5 } { x ^ { 2 } - 12 x + 27 }$

Accepted Answer

x = 3, 6,

Question 15

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 20$ on the interval -9 $\le$ x $\le$ 4.

Accepted Answer

Maximum is 6,419 at

Question 16

=The absolute maximum of the function $f ( x ) = \frac { 2 x } { x + 1 }$ on the interval 0$\le$  x $\le$  1 is 1.

Accepted Answer

A) True 
 B)False

Question 17

Find all critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 4 }$ .

Accepted Answer

The answer of Find all critical numbers of the function...

Question 18

Find all vertical and horizontal asymptotes of the graph of the given function. $$f ( x ) = \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 8 x + 15 }$$

Accepted Answer

A)  Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 0 
B)  Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 1 
C)  Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 2 
D)  Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 1 
A)  Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 0 
B)  Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 1 
C)  Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 2 
D)  Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 1

Question 19

An open box is to be made from a square piece of cardboard, 24 inches by 24 inches, by removing a small square from each corner and folding up the flaps to form the sides. What are the dimensions of the box of greatest volume that can be constructed in this way?

Accepted Answer

A)  8 inches long by 8 inches wide by 4 inches deep 
B)  12 inches long by 12 inches wide by 4 inches deep 
C)  4 inches long by 4 inches wide by 16 inches deep 
D)  16 inches long by 16 inches wide by 4 inches deep 
A)  8 inches long by 8 inches wide by 4 inches deep 
B)  12 inches long by 12 inches wide by 4 inches deep 
C)  4 inches long by 4 inches wide by 16 inches deep 
D)  16 inches long by 16 inches wide by 4 inches deep

Question 20

The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x < \frac { 2 } { 3 }$ .

Accepted Answer

A) True 
 B)False

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 12 x - 5$ .

Find two non-negative numbers x and y whose sum is 4 and are such that the xy² is as large as possible.

Find the absolute maximum and minimum of the function $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval 0 $\le$ x $\le$ 3.

Find two non-negative numbers x and y whose sum is 10 and are such that xy² is as large as possible.

The derivative of function shown below is negative on the interval shown.

Name the vertical and horizontal asymptotes of the given graph.

Determine where the second derivative in the function graphed below is positive.

The function $f ( x ) = x ^ { 6 } - 5 x ^ { 3 } + 2$ has a relative maximum at x = 1.

Find the absolute maximum of the function $f ( x ) = x ^ { 5 } - x ^ { 4 }$ on the interval -1 $\le$ x $\le$ 1.

Find the intervals of increase and decrease for $f ( x ) = 4 x ^ { 3 } + 6 x ^ { 2 } - 72 x - 3$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

Determine where the graph of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ is concave up and concave down.

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\le$ x $\le$ 5.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 5 } { x ^ { 2 } - 12 x + 27 }$

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 20$ on the interval -9 $\le$ x $\le$ 4.

=The absolute maximum of the function $f ( x ) = \frac { 2 x } { x + 1 }$ on the interval 0 $\le$ x $\le$ 1 is 1.

Find all critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 4 }$ .

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 8 x + 15 }$

An open box is to be made from a square piece of cardboard, 24 inches by 24 inches, by removing a small square from each corner and folding up the flaps to form the sides. What are the dimensions of the box of greatest volume that can be constructed in this way?

The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x < \frac { 2 } { 3 }$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

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Exam 3: Additional Applications of the Derivative

Find all the critical numbers of the function f(x)=x3−12x−5f ( x ) = x ^ { 3 } - 12 x - 5f(x)=x3−12x−5 .

Find two non-negative numbers x and y whose sum is 4 and are such that the xy2 is as large as possible.

Find the absolute maximum and minimum of the function f(x)=2x3−7x2+8x+2f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2f(x)=2x3−7x2+8x+2 on the interval 0 ≤\le≤ x ≤\le≤ 3.

Find two non-negative numbers x and y whose sum is 10 and are such that xy2 is as large as possible.

The derivative of function shown below is negative on the interval shown.

Name the vertical and horizontal asymptotes of the given graph.

Determine where the second derivative in the function graphed below is positive.

The function f(x)=x6−5x3+2f ( x ) = x ^ { 6 } - 5 x ^ { 3 } + 2f(x)=x6−5x3+2 has a relative maximum at x = 1.

Find the absolute maximum of the function f(x)=x5−x4f ( x ) = x ^ { 5 } - x ^ { 4 }f(x)=x5−x4 on the interval -1 ≤\le≤ x ≤\le≤ 1.

Find the intervals of increase and decrease for f(x)=4x3+6x2−72x−3f ( x ) = 4 x ^ { 3 } + 6 x ^ { 2 } - 72 x - 3f(x)=4x3+6x2−72x−3 .

Determine where the graph of f(x)=x3−6x2+8x+4f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4f(x)=x3−6x2+8x+4 is concave down.

Determine where the graph of f(x)=x4+6x3−24x2+26f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26f(x)=x4+6x3−24x2+26 is concave up and concave down.

Find the absolute minimum of the function f(x)=x3−3x2f ( x ) = x ^ { 3 } - 3 x ^ { 2 }f(x)=x3−3x2 on the interval 1 ≤\le≤ x ≤\le≤ 5.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. f(x)=5x2−12x+27f ( x ) = \frac { 5 } { x ^ { 2 } - 12 x + 27 }f(x)=x2−12x+275​

Find the absolute maximum and absolute minimum of f(x)=x4−2x2+20f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 20f(x)=x4−2x2+20 on the interval -9 ≤\le≤ x ≤\le≤ 4.

=The absolute maximum of the function f(x)=2xx+1f ( x ) = \frac { 2 x } { x + 1 }f(x)=x+12x​ on the interval 0 ≤\le≤ x ≤\le≤ 1 is 1.

Find all critical numbers of the function f(x)=x3−3x4f ( x ) = x ^ { 3 } - 3 x ^ { 4 }f(x)=x3−3x4 .

Find all vertical and horizontal asymptotes of the graph of the given function. f(x)=x2−2xx2−8x+15f ( x ) = \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 8 x + 15 }f(x)=x2−8x+15x2−2x​

An open box is to be made from a square piece of cardboard, 24 inches by 24 inches, by removing a small square from each corner and folding up the flaps to form the sides. What are the dimensions of the box of greatest volume that can be constructed in this way?

The graph of f(x)=3x4−2x3−12x2+18x−5f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5f(x)=3x4−2x3−12x2+18x−5 is concave up for x<23x < \frac { 2 } { 3 }x<32​ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 12 x - 5$ .

Find two non-negative numbers x and y whose sum is 4 and are such that the xy² is as large as possible.

Find the absolute maximum and minimum of the function $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval 0 $\le$ x $\le$ 3.

Find two non-negative numbers x and y whose sum is 10 and are such that xy² is as large as possible.

The function $f ( x ) = x ^ { 6 } - 5 x ^ { 3 } + 2$ has a relative maximum at x = 1.

Find the absolute maximum of the function $f ( x ) = x ^ { 5 } - x ^ { 4 }$ on the interval -1 $\le$ x $\le$ 1.

Find the intervals of increase and decrease for $f ( x ) = 4 x ^ { 3 } + 6 x ^ { 2 } - 72 x - 3$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

Determine where the graph of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ is concave up and concave down.

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\le$ x $\le$ 5.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 5 } { x ^ { 2 } - 12 x + 27 }$

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 20$ on the interval -9 $\le$ x $\le$ 4.

=The absolute maximum of the function $f ( x ) = \frac { 2 x } { x + 1 }$ on the interval 0 $\le$ x $\le$ 1 is 1.

Find all critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 4 }$ .

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 8 x + 15 }$

The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x < \frac { 2 } { 3 }$ .