MAT180-HY1 Calculus 1

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Let f(x)=(3+4x)4f(x)=(3+4x)4 f(x)f(x) has one critical value at A = Incorrect For x<Ax<A, f(x)f(x) is Correct For x>Ax>A, f(x)f(x) is Correct

Question 1. Last Attempt: 0.7 out of 1 (parts: Incorrect 0/0.33, Correct 0.33/0.33, Correct 0.34/0.34) Score in Gradebook: 0.7 out of 1 (parts: 0/0.33, Correct 0.33/0.33, Correct 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)=−2x3+30x2−54x+10f(x)=-2x3+30x2-54x+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)=2x3−42x2+240x+1f(x)=2x3-42x2+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)=−2x3+33x2−144x+1f(x)=-2x3+33x2-144x+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)=−4x3−13.02x2+459.888x+7.49f(x)=-4x3-13.02x2+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)=12x5+45x4−80x3+1f(x)=12x5+45x4-80x3+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)=−6x3+11.88x2+50.4792x+0.09f(x)=-6x3+11.88x2+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

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)=12x5+75x4−120x3+7f(x)=12x5+75x4-120x3+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)=−4x3+1.26x2+338.8584x+5.33f(x)=-4x3+1.26x2+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)=12x5+60x4−240x3+3f(x)=12x5+60x4-240x3+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

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

Incorrect

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

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)=6x3−27x2+36xg(x)=6x3-27x2+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+6x2−96x+10f(x)=x3+6x2-96x+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)=x4-128x211. (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)=2x3−42x2+240x+10f(x)=2x3-42x2+240x+10 has derivative f'(x)=6x2−84x+240f′(x)=6x2-84x+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)=12x5+75x4−120x3+7f(x)=12x5+75x4-120x3+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 ) = x42−2.5x3−18x2x42-2.5x3-18x2, 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 ) = −x42−0.5x3+42x2-x42-0.5x3+42x2, 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 ) = 122x2+4122x2+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)=6x3+9x2−108xg(x)=6x3+9x2-108x, 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+9x2−21x+3f(x)=x3+9x2-21x+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)=x4-32x29. (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)=12x5+45x4−360x3+4f(x)=12x5+45x4-360x3+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+2.5x3−10.5x2x42+2.5x3-10.5x2, 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 ) = −x42−1.5x3+13.5x2-x42-1.5x3+13.5x2, 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 ) = 193.5x2+1193.5x2+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

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