Math Analysis

 

 

Let f(x)=(3+4x)4f(x)=(3+4x)4

f(x)f(x) has one critical value at A =

For x<Ax<A, f(x)f(x) is

For x>Ax>A, f(x)f(x) is
Question 1. Last Attempt: 0.7 out of 1 (parts: 0/0.33, 0.33/0.33, 0.34/0.34)
Score in Gradebook: 0.7 out of 1 (parts: 0/0.33, 0.33/0.33, 0.34/0.34)
Let f(x)=(7−5x)5f(x)=(7-5x)5

f(x)f(x) has one critical value at A =

For x<Ax<A, f(x)f(x) is

For x>Ax>A, f(x)f(x) is
Question 2. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Let f(x)=(8−5x)7f(x)=(8-5x)7

f(x)f(x) has one critical value at A =

For x<Ax<A, f(x)f(x) is

For x>Ax>A, f(x)f(x) is
Question 3. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=−2×3+30×2−54x+10f(x)=-2×3+30×2-54x+10 has one local minimum and one local maximum.
This function has a local minimum at xx =
with value

and a local maximum at xx =
with value
Question 4. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=2×3−42×2+240x+1f(x)=2×3-42×2+240x+1 has one local minimum and one local maximum.
This function has a local minimum at xx =
with function value

and a local maximum at xx =
with function value
Question 5. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=8x+9x−1f(x)=8x+9x-1 has one local minimum and one local maximum.
This function has a local maximum at x=x=
with value

and a local minimum at x=x=
with value
Question 6. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
For −15≤x≤12-15≤x≤12 the function ff is defined by f(x)=x5(x+6)2f(x)=x5(x+6)2

On which two intervals is the function increasing (enter intervals in ascending order)?
x = to x =
and
x = to x =

Find the interval on which the function is positive: x = to x=

Where does the function achieve its absolute minimum? x =
Question 7. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=−2×3+33×2−144x+1f(x)=-2×3+33×2-144x+1. For this function there are three important open intervals: (−∞,A)(-∞,A), (A,B)(A,B), and (B,∞)(B,∞) where AA and BB are the critical numbers.
Find AA
and BB

For each of the following open intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,∞)(B,∞):
Question 8. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=8x+3x−1f(x)=8x+3x-1. For this function there are four important open intervals: (−∞,A)(-∞,A), (A,B)(A,B),(B,C)(B,C), and (C,∞)(C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following open intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,C)(B,C):
(C,∞)(C,∞):
Question 9. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=−4(x−3)2/3f(x)=-4(x-3)2/3. For this function there are two important open intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where AA is a critical number.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,∞)(A,∞):
Question 10. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e3xf(x)=x2e3x.
For this function there are three important open intervals: (−∞,A)(-∞,A), (A,B)(A,B), and (B,∞)(B,∞) where AA and BB are the critical numbers.
Find AA
and BB

For each of the following intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,∞)(B,∞)
Question 11. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=−4×3−13.02×2+459.888x+7.49f(x)=-4×3-13.02×2+459.888x+7.49
is increasing on the open interval ( , ).

It is decreasing on the open interval ( −∞-∞, ) and the open interval ( , ∞∞ ).

The function has a local maximum at .
Question 12. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=12×5+45×4−80×3+1f(x)=12×5+45×4-80×3+1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B][A,B],[B,C][B,C], and [C,∞)[C,∞) where AA, BB, and CC are the critical numbers.
Find AA
and BB
and CC

At each critical number AA, BB, and CC does f(x)f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER.
At AA
At BB
At CC
Question 13. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=3x+4x−1f(x)=3x+4x-1. For this function there are four important open intervals: (−∞,A)(-∞,A), (A,B)(A,B),(B,C)(B,C), and (C,∞)(C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following open intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,C)(B,C):
(C,∞)(C,∞):
Question 14. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e4xf(x)=x2e4x.
For this function there are three important intervals: (−∞,A](-∞,A], [A,B][A,B], and [B,∞)[B,∞) where AA and BB are the critical numbers.
Find AA
and BB

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B][A,B]:
[B,∞)[B,∞)
Question 15. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=3x+5x−1f(x)=3x+5x-1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B)[A,B),(B,C](B,C], and [C,∞)[C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B)[A,B):
(B,C](B,C]:
[C,∞)[C,∞)
Question 16. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=−6×3+11.88×2+50.4792x+0.09f(x)=-6×3+11.88×2+50.4792x+0.09
is increasing on the open interval ( , ).

It is decreasing on the open interval ( −∞-∞, ) and the open interval ( , ∞∞ ).

The function has a local maximum at .
Question 17. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=3x+4x−1f(x)=3x+4x-1. For this function there are four important open intervals: (−∞,A)(-∞,A), (A,B)(A,B),(B,C)(B,C), and (C,∞)(C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following open intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,C)(B,C):
(C,∞)(C,∞):
Question 18. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e20xf(x)=x2e20x.
For this function there are three important intervals: (−∞,A](-∞,A], [A,B][A,B], and [B,∞)[B,∞) where AA and BB are the critical numbers.
Find AA
and BB

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B][A,B]:
[B,∞)[B,∞)
Question 19. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=3x+2x−1f(x)=3x+2x-1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B)[A,B),(B,C](B,C], and [C,∞)[C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B)[A,B):
(B,C](B,C]:
[C,∞)[C,∞)
Question 20. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=12×5+75×4−120×3+7f(x)=12×5+75×4-120×3+7. For this function there are four important intervals: (−∞,A](-∞,A], [A,B][A,B],[B,C][B,C], and [C,∞)[C,∞) where AA, BB, and CC are the critical numbers.
Find AA
and BB
and CC

At each critical number AA, BB, and CC does f(x)f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER.
At AA
At BB
At CC
Question 21. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=−4×3+1.26×2+338.8584x+5.33f(x)=-4×3+1.26×2+338.8584x+5.33
is increasing on the open interval ( , ).

It is decreasing on the open interval ( −∞-∞, ) and the open interval ( , ∞∞ ).

The function has a local maximum at .
Question 22. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=12×5+60×4−240×3+3f(x)=12×5+60×4-240×3+3. For this function there are four important intervals: (−∞,A](-∞,A], [A,B][A,B],[B,C][B,C], and [C,∞)[C,∞) where AA, BB, and CC are the critical numbers.
Find AA
and BB
and CC

At each critical number AA, BB, and CC does f(x)f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER.
At AA
At BB
At CC
Question 23. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=5x+6x−1f(x)=5x+6x-1. For this function there are four important open intervals: (−∞,A)(-∞,A), (A,B)(A,B),(B,C)(B,C), and (C,∞)(C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following open intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,C)(B,C):
(C,∞)(C,∞):
Question 24. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e7xf(x)=x2e7x.
For this function there are three important intervals: (−∞,A](-∞,A], [A,B][A,B], and [B,∞)[B,∞) where AA and BB are the critical numbers.
Find AA
and BB

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B][A,B]:
[B,∞)[B,∞)
Question 25. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=8x+9x−1f(x)=8x+9x-1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B)[A,B),(B,C](B,C], and [C,∞)[C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B)[A,B):
(B,C](B,C]:
[C,∞)[C,∞)
Question 26. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1

2#

5-1-2-3-4-5

At the point shown on the function above, which of the following is true?
• f'<0,f”<0f′<0,f′′<0
• f’>0,f”>0f′>0,f′′>0
• f’>0,f”<0f′>0,f′′<0
• f'<0,f”>0f′<0,f′′>0

Question 1. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Below is the function f(x)f(x).

1234567-1-2-3-4-5-6-71234567-1-2-3-4-5-6-7

Over which interval of xx values is f’>0f′>0?
• (2,∞)(2,∞)
• [2,∞)[2,∞)
• (−∞,2)(-∞,2)
• (−∞,2](-∞,2]
• (−∞,∞](-∞,∞]

Over which interval of xx values is f'<0f′<0?
• (2,∞)(2,∞)
• [2,∞)[2,∞)
• (−∞,2)(-∞,2)
• (−∞,2](-∞,2]
• (−∞,∞](-∞,∞]

Over the interval (−∞,∞)(-∞,∞), this function is
• concave up (f”>0f′′>0)
• concave down (f”<0f′′<0)
Question 2. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
12345-1-2-3-4-512-1-2

At the point shown on the function above, which of the following is true?
• f’>0,f”<0f′>0,f′′<0
• f’>0,f”>0f′>0,f′′>0
• f'<0,f”>0f′<0,f′′>0
• f'<0,f”<0f′<0,f′′<0
Question 3. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=2x+65x+1f(x)=2x+65x+1. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where the function is not defined at AA.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,∞)(A,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up or concave down.
(−∞,A)(-∞,A):
(A,∞)(A,∞)
Question 4. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=7(x−5)2/3f(x)=7(x-5)2/3. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where AA is a critical number.
AA is

For each of the following intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,∞)(A,∞):

For each of the following intervals, tell whether f(x)f(x) is concave up or concave down.
(−∞,A)(-∞,A):
(A,∞)(A,∞):
Question 5. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=2x+7x−1f(x)=2x+7x-1. For this function there are four important intervals: (−∞,A)(-∞,A), (A,B)(A,B),(B,C)(B,C), and (C,∞)(C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following open intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A):
(A,B)(A,B):
(B,C)(B,C):
(C,∞)(C,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up or concave down.
(−∞,B)(-∞,B):
(B,∞)(B,∞):
Question 6. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e10xf(x)=x2e10x.

f(x)f(x) has two inflection points at x = C and x = D with C < D
where C is
and D is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up or concave down.
(−∞,C)(-∞,C):
(C,D)(C,D):
(D,∞)(D,∞)
Question 7. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
3
Determine the signs (positive, negative, or zero) of y=f(x)y=f(x) (shown in the graph) and f’x)f′x) and f”(x)f′′(x) when x = 3.

The sign of f(3)f(3) is
The sign of f'(3)f′(3) is
The sign of f”(3)f′′(3) is
Question 8. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function g(x)=6×3−27×2+36xg(x)=6×3-27×2+36x, find the first derivative, g'(x)g′(x).
g'(x)=g′(x)=

Notice that g'(x)=0g′(x)=0 when x=2x=2, that is, g'(2)=0g′(2)=0.

Now, we want to know whether there is a local minimum or local maximum at x=2x=2, so we will use the second derivative test.
Find the second derivative, g”(x)(x).
g”(x)=(x)=

Evaluate g”(2)(2).
g”(2)=(2)=

Based on the sign of this number, does this mean the graph of g(x)g(x) is concave up or concave down at x=2x=2?
[Answer either up or down — watch your spelling!!]
At x=2x=2 the graph of g(x)g(x) is concave

Based on the concavity of g(x)g(x) at x=2x=2, does this mean that there is a local minimum or local maximum at x=2x=2?
[Answer either minimum or maximum — watch your spelling!!]
At x=2x=2 there is a local
Question 9. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Let f(x)=x3+6×2−96x+10f(x)=x3+6×2-96x+10.

(a) Use the definition of a derivative or the derivative rules to find
f'(x)=f′(x)=

(b) Use the definition of a derivative or the derivative rules to find
f”(x)=f′′(x)=

(c) On what interval is ff increasing (include the endpoints in the interval)?
interval of increasing =

(d) On what interval is ff decreasing (include the endpoints in the interval)?
interval of decreasing =

(e) On what interval is ff concave downward (include the endpoints in the interval)?
interval of downward concavity =

(f) On what interval is ff concave upward (include the endpoints in the interval)?
interval of upward concavity =
Question 10. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Let f(x)=x4−128x211f(x)=x4-128×211.

(a) Use the definition of a derivative or the derivative rules to find
f'(x)=f′(x)=

(b) Use the definition of a derivative or the derivative rules to find
f”(x)=f′′(x)=

(c) On what interval is ff increasing (include the endpoints in the interval)?
interval of increasing =

(d) On what interval is ff decreasing (include the endpoints in the interval)?
interval of increasing =

(e) On what interval is ff concave downward (include the endpoints in the interval)?
interval of increasing =

(f) On what interval is ff concave upward (include the endpoints in the interval)?
interval of increasing =
Question 11. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
The function f(x)=2×3−42×2+240x+10f(x)=2×3-42×2+240x+10 has derivative f'(x)=6×2−84x+240f′(x)=6×2-84x+240.

Find the critical points, then use the second derivative test to determine whether they are a minimum or a maximum.

f(x) has a local minimum at xx equals

with value

and a local maximum at xx equals

with value
Question 12. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=12×5+75×4−120×3+7f(x)=12×5+75×4-120×3+7.

f(x)f(x) has inflection points at (reading from left to right) x=Dx=D, EE, and FF

where DD is
and EE is
and FF is

For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).

(−∞,D](-∞,D]:
[D,E][D,E]:
[E,F][E,F]:
[F,∞)[F,∞):
Question 13. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=2x+85x+1f(x)=2x+85x+1. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where the function is not defined at AA.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,∞)(A,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,A)(-∞,A):
(A,∞)(A,∞)
Question 14. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=4(x−3)2/3f(x)=4(x-3)2/3. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where AA is a critical number.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,∞)(A,∞):

For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,A)(-∞,A):
(A,∞)(A,∞):
Question 15. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=3x+7x−1f(x)=3x+7x-1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B)[A,B),(B,C](B,C], and [C,∞)[C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B)[A,B):
(B,C](B,C]:
[C,∞)[C,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,B)(-∞,B):
(B,∞)(B,∞):
Question 16. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e8xf(x)=x2e8x.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 17. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
1234567-1-2-3-55-110-165-220-275-330-385-440-495-550
For the above quartic polynomial f( x ) = x42−2.5×3−18x2x42-2.5×3-18×2, identify the interval on which f is concave down.
< x <
Question 18. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
1234567-1-2-3-4-5-6-71032063094125156187218249271030
For the above quartic polynomial f( x ) = −x42−0.5×3+42×2-x42-0.5×3+42×2, identify the interval on which f is concave up.
< x <
Question 19. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
123-1-2-30.30.60.91.21.51.82.12.42.73
For the above rational function f( x ) = 122×2+4122×2+4, identify the interval on which f is concave down.
< x <
Question 20. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
1234-1-2-3-41234-1-2-3-4
For the above rational function f( x ) = 19x4x2+219x4x2+2, identify its three inflection points.
lowest =
middle =
highest =
Question 21. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=e−(x−3)28f(x)=e-(x-3)28.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 22. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=18e−x22f(x)=18e-x22.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 23. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function f(x)=12e16xf(x)=12e16x

List the x-coordinates of the critical values (enter DNE if none)

List the x-coordinates of the inflection points (enter DNE if none)

List the intervals over which the function is increasing or decreasing (use DNE for any empty intervals)
Increasing on
Decreasing on

List the intervals over which the function is concave up or concave down (use DNE for any empty intervals)
Concave up on
Concave down on
Question 24. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function f(x)=x2e3xf(x)=x2e3x,

Determine the open interval(s) where the function is concave up

 

Determine the open interval(s) where the function is concave down

 

Determine any points of inflection.

Question 25. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
(Note: on this question, your answer will initially be marked wrong until the instructor has a chance to grade it; so don’t panic!)

Sketch the graph of a continuous function that satisfies the following conditions:
f'(x)>0f′(x)>0 on (−∞,2)(-∞,2)
f'(x)<0f′(x)<0 on (2,∞)(2,∞)
f”(x)>0f′′(x)>0 on (−1,1)(-1,1)
f”(x)<0f′′(x)<0 on (−∞,−1)(-∞,-1) and (1,∞)(1,∞)
12345-1-2-3-4-512345-1-2-3-4-5
Clear All Draw: Freehand DrawEraser

Question 26. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function g(x)=6×3+9×2−108xg(x)=6×3+9×2-108x, find the first derivative, g'(x)g′(x).
g'(x)=g′(x)=

Notice that g'(x)=0g′(x)=0 when x=−3x=-3, that is, g'(−3)=0g′(-3)=0.

Now, we want to know whether there is a local minimum or local maximum at x=−3x=-3, so we will use the second derivative test.
Find the second derivative, g”(x)(x).
g”(x)=(x)=

Evaluate g”(−3)(-3).
g”(−3)=(-3)=

Based on the sign of this number, does this mean the graph of g(x)g(x) is concave up or concave down at x=−3x=-3?
[Answer either up or down — watch your spelling!!]
At x=−3x=-3 the graph of g(x)g(x) is concave

Based on the concavity of g(x)g(x) at x=−3x=-3, does this mean that there is a local minimum or local maximum at x=−3x=-3?
[Answer either minimum or maximum — watch your spelling!!]
At x=−3x=-3 there is a local
Question 27. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Let f(x)=x3+9×2−21x+3f(x)=x3+9×2-21x+3.

(a) Use the definition of a derivative or the derivative rules to find
f'(x)=f′(x)=

(b) Use the definition of a derivative or the derivative rules to find
f”(x)=f′′(x)=

(c) On what interval is ff increasing (include the endpoints in the interval)?
interval of increasing =

(d) On what interval is ff decreasing (include the endpoints in the interval)?
interval of decreasing =

(e) On what interval is ff concave downward (include the endpoints in the interval)?
interval of downward concavity =

(f) On what interval is ff concave upward (include the endpoints in the interval)?
interval of upward concavity =
Question 28. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Let f(x)=x4−32x29f(x)=x4-32×29.

(a) Use the definition of a derivative or the derivative rules to find
f'(x)=f′(x)=

(b) Use the definition of a derivative or the derivative rules to find
f”(x)=f′′(x)=

(c) On what interval is ff increasing (include the endpoints in the interval)?
interval of increasing =

(d) On what interval is ff decreasing (include the endpoints in the interval)?
interval of increasing =

(e) On what interval is ff concave downward (include the endpoints in the interval)?
interval of increasing =

(f) On what interval is ff concave upward (include the endpoints in the interval)?
interval of increasing =
Question 29. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=12×5+45×4−360×3+4f(x)=12×5+45×4-360×3+4.

f(x)f(x) has inflection points at (reading from left to right) x=Dx=D, EE, and FF

where DD is
and EE is
and FF is

For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).

(−∞,D](-∞,D]:
[D,E][D,E]:
[E,F][E,F]:
[F,∞)[F,∞):
Question 30. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=2x+44x+2f(x)=2x+44x+2. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where the function is not defined at AA.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,∞)(A,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,A)(-∞,A):
(A,∞)(A,∞)
Question 31. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=5(x−2)2/3f(x)=5(x-2)2/3. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where AA is a critical number.
Find AA

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A)(-∞,A):
(A,∞)(A,∞):

For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,A)(-∞,A):
(A,∞)(A,∞):
Question 32. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=2x+3x−1f(x)=2x+3x-1. For this function there are four important intervals: (−∞,A](-∞,A], [A,B)[A,B),(B,C](B,C], and [C,∞)[C,∞) where AA, and CC are the critical numbers and the function is not defined at BB.
Find AA
and BB
and CC

For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A](-∞,A]:
[A,B)[A,B):
(B,C](B,C]:
[C,∞)[C,∞)

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,B)(-∞,B):
(B,∞)(B,∞):
Question 33. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=x2e19xf(x)=x2e19x.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 34. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
12-1-2-3-4-5-6-28-56-84-112-140-168-196-224-252
For the above quartic polynomial f( x ) = x42+2.5×3−10.5x2x42+2.5×3-10.5×2, identify the interval on which f is concave down.
< x <
Question 35. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
123-1-2-3-4-521426384105126147168189210
For the above quartic polynomial f( x ) = −x42−1.5×3+13.5×2-x42-1.5×3+13.5×2, identify the interval on which f is concave up.
< x <
Question 36. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
123-1-2-324681012141618
For the above rational function f( x ) = 193.5×2+1193.5×2+1, identify the interval on which f is concave down.
< x <
Question 37. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
123-1-2-31234-1-2-3-4
For the above rational function f( x ) = 16x4x2+116x4x2+1, identify its three inflection points.
lowest =
middle =
highest =
Question 38. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=e−(x−18)232f(x)=e-(x-18)232.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 39. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Consider the function f(x)=18e−x22f(x)=18e-x22.

f(x)f(x) has two inflection points at x = C and x = D with C≤DC≤D
where CC is
and DD is

Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD).
(−∞,C](-∞,C]:
[C,D][C,D]:
[D,∞)[D,∞)
Question 40. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function f(x)=2e13xf(x)=2e13x

List the x-coordinates of the critical values (enter DNE if none)

List the x-coordinates of the inflection points (enter DNE if none)

List the intervals over which the function is increasing or decreasing (use DNE for any empty intervals)
Increasing on
Decreasing on

List the intervals over which the function is concave up or concave down (use DNE for any empty intervals)
Concave up on
Concave down on
Question 41. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
Given the function f(x)=x2e8xf(x)=x2e8x,

Determine the open interval(s) where the function is concave up

 

Determine the open interval(s) where the function is concave down

 

Determine any points of inflection.

Question 42. Last Attempt: 0 out of 1
Score in Gradebook: 0 out of 1
(Note: on this question, your answer will initially be marked wrong until the instructor has a chance to grade it; so don’t panic!)

Sketch the graph of a continuous function that satisfies the following conditions:
f'(x)>0f′(x)>0 on (−∞,2)(-∞,2)
f'(x)<0f′(x)<0 on (2,∞)(2,∞)
f”(x)>0f′′(x)>0 on (−1,1)(-1,1)
f”(x)<0f′′(x)<0 on (−∞,−1)(-∞,-1) and (1,∞)(1,∞)
12345-1-2-3-4-512345-1-2-3-4-5
Clear All Draw: Freehand DrawEraser

 

 

 

 

Sample Solution

Dissecting How Stigmas and Media Influence Disordered Eating in Males Unique There are different natural, mental, and social factors that add to all conditions or clutters (Fitcher and Krenn, 2003) This writing survey talks about organic, mental, and social factors that impact confused eating, explicitly in guys. Furthermore, this audit underlines the impact that social marks of shame and media have on sentiments of body disappointment which, thus, can be related with side effects of dietary problems. Tragically, the present society ceaselessly uncovered, both, types of people to messages that disclose to them their body is the thing that characterizes them (Duggan and McCreary, 2004). Such messages can be passed on through social variables like ads, disparaged sexual orientation jobs, guardians, and media (Ricciardelli and McCabe, 2004). It has been discovered that introduction to these variables may prompt disguise of the media’s pre-imagined thoughts of the “flawless man”, which can be connected to expanded body disappointment (Thompson and Stick, 2001). For most of guys, muscle-arranged body disappointment prompted a drive for strength (McCreary and Sasse, 2000). Decisively, wild organic elements, sociocultural impacts, and mental unrest are related with body disappointment and symptomatic scattered eating in guys. Investigating How Stigmas and Media Influence Disordered Eating in Males Notwithstanding the expanded predominance of dietary problems among ladies, look into has demonstrated that cluttered eating practices are on the ascent in guys (Furnham, Badmin, and Sneade, 2002). This writing survey separates factors that impact issue in guys into three gatherings: natural, mental, and social elements (Ricciardelli and McCabe, 2004). The biopsychosocial model, clarifies wellbeing and sickness from an organic, mental, and social point of view (Suls and Rothman, 2004). Moreover, this model sees that it isn’t only one framework engaged with the experience and results of a person’s wellbeing or sickness, however each of the three frameworks, intelligently working. (Suls &Rothman, 2004). The impact of social marks of shame and media on a person’s apparent degree of body disappointment shows the effect of social effects on mental working (Harrison and Cantor, 1997). In like manner, wild natural factors, for example, BMI or pubertal planning, may influence levels of body disappointment (Ricciardelli and McCabe, 2004). Lamentably, the present society persistently uncovered the two people to messages that disclose to them their body is the thing that characterizes them (Duggan and McCreary, 2004). Messages, for example, these, can be passed on through guardians, criticized sexual orientation jobs, and media. It has been discovered that a few people may start to disguise, or really accept society’s pre-considered ideas of allure, and start taking part in dietary problem symptomatology to attempt to come to these, to some degree unattainable, objectives (Thompson and Stice, 2001). So how do these organic, mental, and social factors explicitly impact confused eating in guys? Organic Factors of Disordered Eating in Males Physiological variables comprise of the person’s hereditary cosmetics, a segment that one doesn’t really have authority over (Suls and Rothman, 2004). Research has demonstrated that two of the most critical organic components connected to dietary issues among guys are weight list (BMI) and pubertal planning (Ricciardelli and McCabe, 2004). Youthful guys who were overweight, with a higher BMI, announced expanded degrees of body disappointment and cultural weights to diminish fat and increment bulk (Ricciardelli and McCabe, 2004). A person with a higher BMI may experience prodding among friends which could influence the social part of his life. Social disconnection can prompt sentiments of negative effect (wretchedness and nervousness), counting calories, and an improved probability of participating in dietary problem practices (Ricciardelli and McCabe, 2004). Contrasted with ladies, guys hit adolescence around two years after the fact (Fichter and Kreen, 2003). Pubertal planning, is an organic hazard factor of dietary issues in guys in light of the fact that downturn, social separation, and body disappointment may increment contingent upon when pubescence is experienced (Ricciardelli and McCabe, 2004). Guys who were late to develop experienced expanded degrees of body disappointment and were bound to go to be practice subordinate than the individuals who experienced late development (Ricciardelli and McCabe, 2004). Late development additionally expanded side effects of despondency, parental clash, and were seen as less prominent among their friends (Ricciardelli and McCabe, 2004). It is demonstrated that both of these organic factors adversely influenced the person’s mental and social parts of their life, improving the probability of scattered eating. Social Factors of Disordered Eating in Males Messages concerning society’s optimal body are being dispersed to young people by their folks. Research relating to guardians’ job in self-perception aggravations among young ladies is known, yet shouldn’t something be said about the job of guardians in the improvement of the self-perception among young men? McCabe and Ricciardelli (2001) led explore which displayed moms were seen to have a more noteworthy impact for dispositions about self-perception on their children’. Besides, it was seen that through positive remarks, moms were progressively compelling on their child’s self-perception with respect to tolerating abstaining from excessive food intake as a way to get in shape (Wertheim, Martin, Prior, Sanson, and Smart, 2002).. Then again, through analysis, fathers were additionally tolerating of activity as a procedure for weight reduction (McCabe and Ricciardelli., 2001) Studies propose that media patterns might be connected to the advancement of dietary issues in media buyers. In the present Western culture male’s physical uncertainties are focused by stressing the requirement for extraordinary weight control plans, improving enhancements, or work out regimes. Notwithstanding the item being sold, the message that men constantly need to improve themselves is being publicized in magazines, for example, Men’s Health. (Duggan and McCreary, 2004). Usually pictures in the media add to body disappointment in ladies at the same time, Pope et al. (2000) has carried another point of view to the table concerning guys. Research recommends that men may have it more terrible than ladies as far as self-perfect inconsistency (Duggan and McCreary, 2004). Duggan and McCreary (2014) accept that following quite a while of commercials with respect to appearance, ladies have figured out how to overlook or face the media (Duggan and McCreary, 2004). In any case, men are seen to be molded to socially refrain from talking about such issues and think that its awkward to express their physical frailties (Duggan and McCreary, 2004). This finding is a ramifications that men are more vulnerable to average impacts than ladies regarding body disappointment and therefore, they will in general purchase publicized items to upgrade their physical appearance (Pope et al, 2000). Marks of shame related with sexual orientation jobs are a noteworthy supporter of self-perception unsettling influences in guys. The manliness theory suggests that men are in danger for strength situated body disappointment and confused eating because of sex jobs including predominance, certainty, sexual achievement, and physical and enthusiastic poise (Griffiths et al., 2014). Results from this examination demonstrated that more noteworthy muscle disappointment and strength situated disarranged eating, yet not muscle to fat ratio disappointment or slenderness arranged scattered eating, was related with expanded adjustment to manly standards. A conceivable clarification for this is a few guys feel as though they come up short on these manly characteristics or are progressively saved, subsequently they feel as though a bigger, increasingly strong body could make up for those emotions prompting body disappointment and cluttered eating. (Griffiths et al., 2014) So how social factors and messages from the media effect communicate with mental working of a person? Mental Factors of Disordered Eating in Males As found in the past areas, organic and social variables are connected to negative affectivity, self-perception disguise, and body disappointment. Self-perception is the manner by which one sees and assesses their appearance and physical capacity (Taylor, 2015). Research has demonstrated that various relationship of self-perception are diverse in men than ladies. Guys are bound to connect engaging quality with expanded, fit, muscle definition, driving them to a drive for strength (McCreary and Sasse, 2000). Then again, females with body disappointment ordinarily partner engaging quality with being meager. (Duggan and McCreary, 2004). Media patterns might be connected to disguise which, thus, can prompt the advancement of dietary problems (Harrison and Cantor, 1997). People disguise the perfect self-perception that the media depicts due to slandered sex jobs. This idea, self-perception disguise, alludes to how much an individual truly accepts society’s meaning of appeal and takes part in practices, for example, disarranged eating, to attempt to arrive at these to some degree unattainable thoughts. Hence, it is the disappointment of the body that hypothetically advances eating less junk food and negative effect, which can expand the hazard for beginning of eating scattered indications. (Thompson and Stice, 2001). Dietary issues in Males There are numerous kinds of dietary issues, nonetheless, the most widely recognized are anorexia nervosa, fanatically eating fewer carbs as well as practicing until one arrives at a body weight very under ideal level, and bulimia, which includes rotating cycles of gorging then vomiting utilizing strategies, for example, regurgitating, intestinal medicines, outrageous consuming less calories, and medication or liquor misuse (Taylor, 2015). Research has demonstrated that there are significantly higher rates of eating disor

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