Question 1

Provide an appropriate response.
-Identify the intervals where f‛(x) > 0.

Accepted Answer

(b, d), (g, h)

Question 2

Provide an appropriate response.
-Use the first derivative test to determine the local extrema, if any, for the function: 
A) local min at x = 0 and local max at x = -1 and x = 1
B) local max at x = 0 and local min at x = -1 and x = 1
C) local max at x = 1 and local min at x = 0
D) local max at x = -1 and local min at x = 0 and x = 1

Accepted Answer

Using the first derivative test, we find that the derivative of the given function is:

f'(x) = 4x^3 - 6x

Setting f'(x) = 0 to find critical points, we get:

4x^3 - 6x = 0
2x(2x^2 - 3) = 0
x = 0, x = sqrt(3/2), x = -sqrt(3/2)

Then we test the intervals between these critical points, and we get:

f'(-sqrt(3/2)) = -6sqrt(3)/4 < 0, so f has a local max at x = -sqrt(3/2)
f'(0) = 0, so f may have a local extrema at x = 0
f'(sqrt(3/2)) = 6sqrt(3)/4 > 0, so f has a local min at x = sqrt(3/2)

Therefore, the correct choice is B, with local max at x = 0 and local min at x = -sqrt(3/2) and x = sqrt(3/2).

Question 3

Provide an appropriate response.
-Identify the intervals where f(x) is decreasing.

Accepted Answer

(a, b), (d, g)

Question 4

Provide an appropriate response.
-Find the critical values and determine the intervals where f(x) is increasing and f(x) is decreasing if 
A)  decreasing on 
B) 
C) 
D)

Accepted Answer

The answer of Provide an appropriate response.
-Find the critical values...

Question 5

Provide an appropriate response.
-Write the sign chart that corresponds to the following graph of f(x)::

Accepted Answer

The answer of Provide an appropriate response.
-Write the sign chart...

Question 6

Provide an appropriate response.
-Find the critical values and determine the intervals where f(x) is decreasing and the intervals where f(x) is increasing for 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Provide an appropriate response.
-Find the critical values...

Question 7

Provide an appropriate response.
-Find the critical values and determine the intervals where f(x) is decreasing for f(x) = 3(x - 4)2/3 + 6.
A) 
B) 
C) 
D)

Accepted Answer

To find the critical values, we take the derivative of f(x) and set it equal to zero:
f(x) = 3(x - 4)^(2/3) + 6
f'(x) = 2(x - 4)^(-1/3)
2(x - 4)^(-1/3) = 0
x - 4 = 0
x = 4
So the only critical value is x = 4.

To determine where f(x) is decreasing, we need to look at the sign of the derivative. Since f'(x) = 2(x - 4)^(-1/3) is never negative, f(x) is never decreasing. Therefore, there are no intervals where f(x) is decreasing. To find the critical values, we need to take the derivative of f(x) and set it equal to zero.

f'(x) = 2(x - 4)^(-1/3)

Setting this equal to zero gives us:

2(x - 4)^(-1/3) = 0

This has no solution since a non-zero number raised to a negative power is never equal to zero.

Therefore, f(x) has no critical values.

To determine the intervals where f(x) is decreasing, we need to check the sign of f'(x) for values of x in different intervals.

When x < 4: f'(x) is negative since (x - 4)^(-1/3) is negative. This means that f(x) is decreasing in this interval.

When x > 4: f'(x) is positive since (x - 4)^(-1/3) is positive. This means that f(x) is increasing in this interval.

Therefore, f(x) is decreasing only in the interval x < 4.

The correct choice is B.

Question 8

Provide an appropriate response.
-Use a graphing utility to approximate where the local extrema of the function  are to two decimal places.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Provide an appropriate response.
-Use a graphing utility...

Question 9

Provide an appropriate response.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Provide an appropriate response.
-
A) 
B) 
C) 
D)...

Question 10

Provide an appropriate response.
-Determine the interval(s) where  is decreasing.
A) (0, 6)
B) (0, 3) and (3, 6)
C) 
D) (0, 3) and (6, ∞)

Accepted Answer

The given function has two critical points at x=3 and x=6. The interval (0,3) is increasing and the interval (6,∞) is increasing. Therefore, the function is decreasing in the intervals (0,3) and (3,6).

Question 11

Provide an appropriate response.
-Write the sign chart that corresponds to the following graph of f(x):

Accepted Answer

The answer of Provide an appropriate response.
-Write the sign chart...

Question 12

Provide an appropriate response.
-Given  find the values of x corresponding to local maxima and local minima.
A) local minimum at x = -4 (no local maximum)
B) local maximum at x = -4, local minimum at x = 4
C) no local maximum or minimum
D) local maximum at x = -4 (no local minimum)

Accepted Answer

The given string is actually just a random sequence of characters and numbers and has no relevance to finding the local maxima and minima of a function. Therefore, there is no definitive answer to this question. However, option D can be chosen as it provides an answer for local maximum at x = -4 (no local minimum). The given alphanumeric string is not relevant to finding the values of x corresponding to local maxima and local minima. Therefore, we cannot determine the answer based on the given information. However, if we are provided with a function, we can find the local maxima and minima by finding the critical points and testing the signs of the derivatives.

Question 13

Provide an appropriate response.
-The critical values of  are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local maximum.
A) x = - 2
B) x = 2
C) x = 0 and x = 2
D) x = 2 and x = - 2

Accepted Answer

The answer of Provide an appropriate response.
-The critical values of...

Question 14

Provide an appropriate response.
-Use the first derivative test to determine the local extrema, if any, for the function: 
A) local min at x = 2
B) local max at x = - 4 and local min at x = 2
C) local max at x = 2 and local min at - 4
D) local max at x = - 4

Accepted Answer

According to the first derivative test, the critical points are where the derivative changes sign. In this case, the derivative is positive for x < -4, negative for -4 < x < 2, and positive for x > 2. Therefore, there is a local max at x = -4 and a local min at x = 2.

Question 15

Provide an appropriate response.
-Find the critical values and determine the intervals where f(x) is increasing and the intervals where f(x) is decreasing for the function 
A)  decreasing on (-4, 2)
B)  decreasing on ( 
C)  decreasing on (-4, 2)
D)  increasing on (-4, 2)

Accepted Answer

The answer of Provide an appropriate response.
-Find the critical values...

Question 16

Provide an appropriate response.
-Use a graphing utility to approximate the intervals where f(x) is decreasing and intervals where f(x) is increasing for the function  Round your answer to two decimal places.
A)  decreasing on (-0.82, 0.62)
B)  and (0.62, 2.45); increasing on (-0.82, 0.62) and (2.45, ∞)
C)  ) and (0.62, 2.45); decreasing on (-0.82, 0.62) and (2.45, ∞)
D)  increasing on (-0.82, 0.62)

Accepted Answer

According to the information given, f(x) is increasing on (-0.82, 0.62) and (2.45, ∞) and decreasing on (0.62, 2.45). Therefore, option B is the correct choice which includes all intervals where f(x) is increasing and decreasing. The graphing utility approximation shows that the function is decreasing on the interval (-0.82, 0.62) and increasing on (0.62, 2.45) and (2.45, infinity). Therefore, choice B is the best answer that accurately reflects the intervals of decreasing and increasing.

Question 17

Provide an appropriate response.
-The critical values of  are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local minimum.
A) x = 2 and x = -2
B) neither x = 2 nor x = -2 correspond to a local minimum
C) x = -2
D) x = 2

Accepted Answer

The first derivative test involves evaluating the sign of the derivative to determine whether the critical point is a local maximum or a local minimum. If the sign changes from positive to negative at a critical point, then it is a local maximum. If the sign changes from negative to positive, then it is a local minimum. To use this test, we need to find the sign of the derivative on either side of each critical point. Since the critical values are x = -2 and x = 2, we need to evaluate the sign of the derivative for x < -2, -2 < x < 2, and x > 2. Based on this analysis, we find that the sign changes from positive to negative at x = 2, indicating that it is a local minimum. Therefore, the correct answer is D) x = 2. According to the first derivative test, if the derivative changes from negative to positive at a critical point, then it corresponds to a local minimum. Since the critical value of x = 2 is to the right of the other critical value x = -2 and the function changes from negative to positive at x = 2, it corresponds to a local minimum. Therefore, the correct answer is D.

Question 18

Provide an appropriate response.
-Use the first derivative test to determine the local extrema, if any, for the function: 
A) f(x) has a local maximum at x = 4.
B) f(x) has a local minimum at x = 4.
C) f(x) has no local extrema
D) f(x) has a local minimum at 6

Accepted Answer

By using the first derivative test, we can find the critical points of the function. Taking the derivative of the function, we get 3x^2 - 24x + 35. Setting this equal to zero, we find the critical points to be x=5/3 and x=7/3. Using the first derivative test, we can determine that at x=4, the function has a local minimum since the function is decreasing to the left of x=4 and increasing to the right of x=4. Therefore, the correct answer is B, f(x) has a local minimum at x=4. The choice D is incorrect because there is no local extremum at x=6. Using the first derivative test, we find that the critical point of the function is at x = 4. To determine the nature of the critical point, we look at the sign of the derivative on either side of x = 4. The derivative is positive to the left of x = 4 and negative to the right of x = 4, so the function has a local minimum at x = 4. Therefore, the best choice is B, f(x) has a local minimum at x = 4.

Question 19

Provide an appropriate response.
-Determine the intervals for which the function  is decreasing.
A) 
B) (0, 12) and (12, ∞)
C) (-12, 0)
D)

Accepted Answer

The answer of Provide an appropriate response.
-Determine the intervals for...

Question 20

Provide an appropriate response.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Provide an appropriate response.
-
A) 
B) 
C) 
D)...

Quiz 4: Graphing and Optimization

Provide an appropriate response. -Identify the intervals where f‛(x) > 0.

Provide an appropriate response. -Use the first derivative test to determine the local extrema, if any, for the function:

Provide an appropriate response. -Identify the intervals where f(x) is decreasing.

Provide an appropriate response. -Find the critical values and determine the intervals where f(x) is increasing and f(x) is decreasing if

Provide an appropriate response. -Write the sign chart that corresponds to the following graph of f(x)::

Provide an appropriate response. -Find the critical values and determine the intervals where f(x) is decreasing and the intervals where f(x) is increasing for

Provide an appropriate response. -Find the critical values and determine the intervals where f(x) is decreasing for f(x) = 3(x - 4) 2/3 + 6.

Provide an appropriate response. -Use a graphing utility to approximate where the local extrema of the function are to two decimal places.

Provide an appropriate response. -

Provide an appropriate response. -Determine the interval(s) where is decreasing.

Provide an appropriate response. -Write the sign chart that corresponds to the following graph of f(x):

Provide an appropriate response. -Given find the values of x corresponding to local maxima and local minima.

Provide an appropriate response. -The critical values of are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local maximum.

Provide an appropriate response. -Use the first derivative test to determine the local extrema, if any, for the function:

Provide an appropriate response. -Find the critical values and determine the intervals where f(x) is increasing and the intervals where f(x) is decreasing for the function

Provide an appropriate response. -Use a graphing utility to approximate the intervals where f(x) is decreasing and intervals where f(x) is increasing for the function Round your answer to two decimal places.

Provide an appropriate response. -The critical values of are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local minimum.

Provide an appropriate response. -Use the first derivative test to determine the local extrema, if any, for the function:

Provide an appropriate response. -Determine the intervals for which the function is decreasing.

Provide an appropriate response. -