The given series are in conditionally convergent
To determine whether the given series is absolutely convergent, conditionally convergent, or divergent, we will use the Comparison Test.
Series in question:
∑ [[tex]5(k(-1)^k + 1)] / (k^2),[/tex] k = 1 to ∞
Step 1: Find the absolute value of the series
| 5([tex]k(-1)^k + 1) / k^2[/tex] |
Step 2: Simplify the absolute value
[tex]5(k + (-1)^k) / k^2[/tex]
Step 3: Use the Comparison Test
We will compare this series to the series ∑ 5k / [tex]k^2,[/tex] k = 1 to ∞.
Since [tex](-1)^k[/tex] is always either 1 or -1, we know that [tex]5(k + (-1)^k) / k^2 \leq 5k / k^2.[/tex]
Step 4: Determine if the comparison series converges
The comparison series can be simplified as
∑ 5 / k, k = 1 to ∞, which is a harmonic series that is known to be divergent.
Step 5: Determine the original series' convergence status
Since the comparison series is divergent, we cannot determine if the original series is absolutely convergent using the Comparison Test.
However, we can now investigate if the series is conditionally convergent by considering the alternating series
∑ (-1)^k(5k) / [tex]k^2[/tex], k = 1 to ∞.
Since the series' terms decrease in magnitude (5k / [tex]k^2[/tex] decreases as k increases) and the limit of the terms as k approaches infinity is zero, the series is conditionally convergent by the Alternating Series Test.
In conclusion, the given series is conditionally convergent.
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Qn in attachment
.
..
Answer:
option d
Step-by-step explanation:
24
pls mrk me brainliest (*￣(ｴ)￣*)
If a bag of marbles contains 6 yellow, 8 blue, and 6 red marbles, then what is the probability of not pulling out a
blue or yellow marble?
Step-by-step explanation:
a probability is always the ratio
desired cases / totally possible cases.
we have here a total of 6 + 8 + 6 = 20 marbles.
to not pull a blue or yellow marble is in this context the same event as pulling a red marble.
so, the desired cases are 6 (red).
which we can get directly from the 6 red marbles, or by counting off the undesired cases : 20 - 8 - 6 = 6.
and the probabilty for not pulling a blue or yellow marble (or simply pulling a red marble) is
6/20 = 3/10 = 0.3
What three-dimensional figure is formed when the triangle shown is rotated around the dashed line?
A. cone
B. cylinder
C. double cone
D. hemisphere
Answer: C
Step-by-step explanation: after rotating, if you split it in half horizontally, you have two cones
The three-dimensional figure formed when the triangle is rotated around the dashed line through B and C is a cone.
What is a cone?A cone is a three-dimensional geometric form with a flat base and a smooth, tapering apex or vertex. A cone is made up of a collection of line segments, half-lines, or lines that link the base's points to the apex, which is a common point on a plane that does not include the base.
When we rotate a two-dimensional shape around an axis, we create a three-dimensional solid. This process is known as "revolution" or "rotational symmetry".
In this particular case, we have a triangle that can be rotated around the line segment that connects points B and C. If we were to rotate the triangle around this axis, we would create a three-dimensional solid. To figure out what kind of solid this is, we can think about the cross-sections that would be created if we were to slice through the solid perpendicular to the axis of rotation.
If we were to slice through the solid perpendicular to the axis of rotation, we would get a circle. This means that the solid created by rotating the triangle is a cylinder.
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What is the value of the expression below? (3 1/2 - 9 3/4) entre (-2.5)
PLEASE HELP
Answer:
Let's solve this in steps:
1. Convert mixed numbers to fractions:
```
3 1/2 = 7/2
9 3/4 = 39/4
```
2. Perform the subtraction:
```
7/2 - 39/4 = -11/4
```
3. Divide by -2.5:
```
-11/4 / -2.5 = 4.4
```
Therefore, the value of the expression is **4.4**.
Find the values of x and y that make the quadrilateral a parallelogram
DEFG
5x-4
3y+9
10x-24
2y+16
Answer:
x = 4, y = 7
Step-by-step explanation:
3y + 9 = 2y + 16
y = 7
10x - 24 = 5x - 4
5x = 20
x = 4
values of x is 4 and y is 7
6. Yu is considering two different banks for his $3,000 savings account: OPTION A 4% FOR 20 YEARS SIMPLE INTEREST OPTION B 2% FOR 10 YEARS COMPOUND INTEREST 8 What is the interest earned on option A? O What is the total value on option A? What is the interest earned on option B? O What is the total value on option B? 10 Which is the better option?
The interest earned on Option A is $2400 and Option B is $666.18. Option A is the better option as Option A has a higher total value of $5400 compared to Option B's total value of $3666.18.
To calculate the interest earned and total value for each option, we can use the following formulas:
For Option A:
- Interest earned = principal x rate x time = 3000 x 0.04 x 20 = $2400
- Total value = principal + interest earned = 3000 + 2400 = $5400
For Option B:
- Interest earned = principal x (1 + rate/n)^(n x time) - principal = 3000 x (1 + 0.02/1)^(1 x 10) - 3000 = $666.18
- Total value = principal + interest earned = 3000 + 666.18 = $3666.18
Therefore, the interest earned and total value for each option are as follows:
Option A:
- Interest earned = $2400
- Total value = $5400
Option B:
- Interest earned = $666.18
- Total value = $3666.18
To compare the two options, we need to consider the total value of each option. Option A has a higher total value of $5400 compared to Option B's total value of $3666.18. Therefore, Option A is the better option.
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Triangle XYZ undergoes a transformation to produce triangle XYZ. The coordinates of both triangles are shown.
X'(6,-1)
X(6, 1)
Y(3,4) Y'(3.-4)
Z(-2,0)→ Z'(-2,0)
Which of the following best describes the transformation?
The transformation of the triangle is reflection over the x-axis
Given data ,
Let the transformation be represented as A
Now , the triangle is given as XYZ
where the coordinates are X ( 6 , 1 ) , Y ( 2 , 4 ) and Z ( -2 , 0 )
Now , the coordinates of the transformed triangle is
X' ( 6 , -1 ) , Y' ( 3 , -4 ) and Z' ( -2 , 0 )
The reflection of point (x, y) across the x-axis is (x, -y)
Hence , the transformation is reflection over x-axis
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if you pay $ for a 20-year zero coupon bond with a face value of $, what is your annual compound rate of return?
The annual compound rate of return on this 20-year zero coupon bond is 6%. To calculate the annual compound rate of return, we need to use the following formula:
Annual Compound Rate of Return = (Face Value / Purchase Price)^(1/Number of Years) - 1
Here, the face value of the bond is $1000, the purchase price is $500, and the bond has a term of 20 years. Substituting these values in the above formula, we get:
Annual Compound Rate of Return = (1000/500)^(1/20) - 1
Simplifying this expression, we get:
Annual Compound Rate of Return = 1.06 - 1
Annual Compound Rate of Return = 0.06 or 6%
Therefore, the annual compound rate of return on this 20-year zero coupon bond is 6%.
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CAN somebody pl help
The expression 8(4 - π) yd² is the area of the of the shaded region in terms of π.
How to evaluate for the area of the shaded regionThe area of the shaded region is the area of the semicircle subtracted from the area of the rectangle
radius of the semicircle is also the width of the rectangle, so;
area of the rectangle = 8 yd × 4 yd = 32 yd²
area of the semicircle = (π × 4 yd × 4 yd)/2
area of the semicircle = 8π yd²
area of the shaded region = 32 yd² - 8π yd²
area of the shaded region = 8(4 - π) yd²
Therefore, the expression 8(4 - π) yd² is the area of the of the shaded region in terms of π.
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WHATS THE AREA PLEASE HELP DUE in 5 minutes
Answer:
The answer to your problem is, 201.06 or 201.1
Step-by-step explanation:
To find the area you use the formula:
A = π [tex]r^2[/tex]
R = Radius
A = Area
We know the radius of the circle is 8
So replace A = π [tex]r^2[/tex]
= π × 8 ≈ 201.06193
Or 201.06 or 201.1
Thus the answer to your problem is, 201.06 or 201.1
Tyler rides his bike from his house to his cousin's house. He bikes a total of 1.8 kilometers to get there and back. What is the distance, in meters, between Tyler's house and his cousin's house?
Answer:
Step-by-step explanation:
Total of rides from Tyler's house to Cousin's house and Cousin's house to Tyler's house = 1.8km = 1800m
So, the distance from Tyler's house to his cousin's house is
= 1800m ÷ 2 = 900m
A study is designed to test the hypotheses h0: m $ 26 versus ha: m , 26. a random sample of 50 units was selected from a specified population, and the measurements were summarized to y 5 25.9 and s 5 7.6. a. with a 5 .05, is there substantial evidence that the population mean is less than 26
The p-value for a t-score of -0.92 is approximately 0.18 and since it is greater than the significant level, the null hypothesis is rejected.
The first step in testing this hypothesis is to calculate the test statistic, which in this case is a t-score. The formula for the t-score is (y - mu) / (s / sqrt(n)), where y is the sample mean, mu is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
In this case, the sample mean is 25.9, the hypothesized population mean is 26, the sample standard deviation is 7.6, and the sample size is 50. Plugging these values into the formula, we get a t-score of -0.92.
Next, we need to find the p-value associated with this t-score. We can use a t-table or a calculator to do this. Using a t-table with 49 degrees of freedom (since we have a sample size of 50 and one parameter estimated from the sample), we find that the p-value for a t-score of -0.92 is approximately 0.18.
Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. In other words, we do not have substantial evidence to conclude that the population mean is less than 26. However, it is important to note that the sample mean is slightly below the hypothesized population mean, and the p-value is relatively close to the significance level. Therefore, it may be worthwhile to conduct additional studies with larger sample sizes or different populations to further investigate this question.
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Which equation represents a line that is perpendicular to the line
represented by 2x - y = 7?
(1) y = -x + 6
(2) y = x + 6
(3) y = -2x + 6
(4) y = 2x + 6
7*. All lengths are in cm. Find the area of the right angled
triangle.
x-14( shortest side)
2x+5( hypotenuse)
2x+3( remaining side)
Answer:
504 cm^2.
Step-by-step explanation:
By Pythagoras:
(2x + 5)^2 = (2x + 3)^2 + (x - 14)^2
4x^2 + 20x + 25 = 4x^2 + 12x + 9 + x^2 - 28x + 196
20x - 12x + 28x + 25 - 9 - 196 = x^2
x^2 - 36x + 180 = 0
(x - 6)(x - 30) = 0
x = 6, 30.
As one of the sides is x - 14, x mst be 30 as its length has to be positive.
So the area of the triangle
= 1/2 * (x - 14) 8 (2x + 3)
= 1/2 * (30-14)(60 + 3)
= 1/2 * 16 * 63
= 504 cm^2.
An engineer is using computer-aided design (CAD) software to design a component for a space shuttle. The scale of the drawing is 1 cm: 60 in. The actual length of the component is 12. 75 feet. What is the length of the component in the drawing?
The length of the component in the drawing is 2.125 centimeters.
How to find the length of the component represented in a CAD?To find the length of the component in the drawing, we convert the given length from feet to inches. Since 1 foot is equal to 12 inches, the actual length of 12.75 feet is equivalent to 12.75 x 12 = 153 inches.
Next, we apply the scale of the drawing, which is 1 cm: 60 in. This means that for every 60 inches in reality, the drawing represents it as 1 centimeter. To find the length in centimeters, we set up a proportion:
1 cm / 60 in = x cm / 153 in
Cross-multiplying and solving for x, we get:
x = (1 cm * 153 in) / 60 in = 2.55 cm
Rounding to three decimal places, the length of the component in the drawing is approximately 2.125 centimeters.
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3. Compute the integral JSS, udv, where U is the part of the ball of radius 3, centered at 0,0,0), that lies in the 1st octant. Recall that the first octant is the part of the 3d space where all three coordinates I, y, z are nonnegative. (Hint: You may use cylindrical or spherical coordinates for this computation, but note that the computation with cylindrical coordinates will involve a trigonometric substitution - 30 spherical cooridnates should be preferable.)
To compute the integral JSS, udv, where U is the part of the ball of radius 3, centered at 0,0,0), that lies in the 1st octant, we can use spherical coordinates. Since the region is defined as having all three coordinates nonnegative, we can set our limits of integration as follows: 0 ≤ ρ ≤ 3, 0 ≤ θ ≤ π/2, and 0 ≤ φ ≤ π/2.
Using the Jacobian transformation, we have:
JSS, udv = ∫∫∫U ρ²sinφ dρdθdφ
Substituting in our limits of integration, we get:
JSS, udv = ∫0^π/2 ∫0^π/2 ∫0³ ρ²sinφ dρdθdφ
Evaluating the integral, we get:
JSS, udv = (3³/3) [(sin(π/2) - sin(0))] [(1/2) (π/2 - 0)]
JSS, udv = 9/2 π
Therefore, the value of the integral JSS, udv, over the part of the ball of radius 3 that lies in the 1st octant is 9/2π.
To compute the integral JSS, udv, over the region U, which is the part of the ball of radius 3 centered at (0,0,0) and lies in the first octant, we will use spherical coordinates for this computation as it's more preferable.
In spherical coordinates, the volume element is given by dv = ρ² * sin(φ) * dρ * dφ * dθ, where ρ is the radial distance, φ is the polar angle (between 0 and π/2 for the first octant), and θ is the azimuthal angle (between 0 and π/2 for the first octant).
Now, we need to set up the integral for the volume of the region U:
JSS, udv = ∫∫∫ (ρ² * sin(φ) * dρ * dφ * dθ), with limits of integration as follows:
ρ: 0 to 3 (radius of the ball),
φ: 0 to π/2 (for the first octant),
θ: 0 to π/2 (for the first octant).
So, the integral becomes:
JSS, udv = ∫(0 to π/2) ∫(0 to π/2) ∫(0 to 3) (ρ² * sin(φ) * dρ * dφ * dθ)
By evaluating this integral, we will obtain the volume of the region U in the first octant.
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A garden hose can normally fill a child's inflatable pool in 30 minutes.
The pool has a small hole in it, and water is secretly leaking out. This leak could empty the
pool in two hours (120 minutes).
How long would it take, from start to finish, until the pool is full of water?
2a) Clearly write out the equation you would use to answer the question.
2b) Answer the question. How long would it take? Please write your answer as a
complete sentence with appropriate units.
2a) The equation used to answer the question is (1/Time to fill the pool) = (1/Time taken by hose) - (1/Time taken by leak).
2b) It would take 40 minutes to fill the pool with water when there is a small hole causing a leak.
To solve this, we can use the concept of rates of work.
2a) The equation we would use to answer the question is:
(1/Time to fill the pool) = (1/Time taken by hose) - (1/Time taken by leak)
2b) Let's plug in the values given in the question:
(1/Time to fill the pool) = (1/30 minutes) - (1/120 minutes)
To find the time to fill the pool, we first need to find a common denominator for the fractions. The common denominator is 120, so we can rewrite the fractions as:
(1/Time to fill the pool) = (4/120) - (1/120)
Now, add the fractions on the right side:
(1/Time to fill the pool) = (3/120)
Next, take the reciprocal of both sides to solve for the time to fill the pool:
Time to fill the pool = 120/3
Time to fill the pool = 40 minutes
So, it would take 40 minutes to fill the pool.
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From monday through friday, earl works in the bookstore on 1 and in the athletic center on another 2 days. on saturday and sunday, earl cooks food 50% of the days. how many days does earl work in a week? what percent of monday through friday does earl work?
Earl works a total of 3 days in a week. From Monday through Friday, he works in the bookstore on 1 day and in the athletic center on 2 days. On Saturday and Sunday, he cooks food on 50% of the days, which would be a total of 1 day. Therefore, he works a total of 3 days in a week.
To calculate the percentage of Monday through Friday that Earl works, we need to first calculate the total number of days in a week, which is 7. Then, we need to subtract the weekend days, which are Saturday and Sunday, leaving us with 5 days.
Finally, we can calculate the percentage by dividing the number of days Earl works from Monday through Friday (which is 1) by the total number of weekdays (which is 5), and multiplying by 100. So, Earl works 20% of Monday through Friday.
In summary, Earl works 3 days in a week, 1 day in the bookstore and 2 days in the athletic center. He also cooks food on 1 day during the weekend. Earl works 20% of Monday through Friday.
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Identify the fractions between 1/16 and 5/8
The fractions between 1/16 and 5/8 are 3/16 and 5/16
Identifying the fractions between 1/16 and 5/8The fraction expressions are given as
1/16 and 5/8
The above fractions are proper fractions because numerator < denominator
Express the fraction 5/8 as a denominator of 16
So, we have the following equivalent fractions
1/16 and 10/16
This means that the fractions between 1/16 and 5/8 can be represented as
a/16
Where
1 < a < 10
So, we have
Possible fraction = 3/16 and 5/16
Hence, the fractions between 1/16 and 5/8 are 3/16 and 5/16
Note that there are other possible fractions too
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Which of the following statements proves the series –128 + 96 – 72 + 54 – … is geometric? r equals negative three fourths r equals three fourths r equals negative four thirds r equals four thirds
Answer: To determine if the series –128 + 96 – 72 + 54 – ... is a geometric series, we need to check if the ratio between consecutive terms is constant.
Let's calculate the ratio between the second and first terms:
96 / (-128) = -3/4
Now let's calculate the ratio between the third and second terms:
-72 / 96 = -3/4
The ratio between the fourth and third terms is:
54 / (-72) = -3/4
We can see that the ratio between consecutive terms is always the same: -3/4. Therefore, the series –128 + 96 – 72 + 54 – ... is a geometric series with a common ratio of -3/4.
So the answer is r equals negative three fourths.
Step-by-step explanation:
The series provided is a geometric series because each term after the first is found by multiplying the previous term by -3/4. Therefore, the common ratio 'r' equals -3/4.
Explanation:In a geometric series, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the series given –128 + 96 – 72 + 54 – …, the second term (96) divided by the first term (-128) equals -3/4, the third term (-72) divided by the second term (96) also equals -3/4, and so on. This constant ratio between successive terms demonstrates that this is indeed a geometric series. Therefore, the statement that proves this is a geometric series is 'r equals negative three fourths' where r represents the common ratio.
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The area of triangle ABC is 4 root 2. Work out the value of x
Question is from mathswatch
$3,900 at 1% compounded
annually for 6 years
_____________________________
A = P (1 + 1%) n = 3,900 (1 + 1%) ⁶= $4,139.92_____________________________
bacteria in a dirty glass triple every day. if there are 25 bacteria to start, how many are in the glass after 15 days
Answer:
Step-by-step explanation:
25x3x15
Use the given acceleration function and initial conditions to find the velocity vector v(t), and position vector r(t). Then find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) =
find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) = This gives you the position vector r(9) as a function of sin(9) and cos(9).
To find the velocity vector v(t) and position vector r(t), we need to integrate the given acceleration function a(t) and apply the initial conditions. Here's a step-by-step explanation:
1. Given acceleration function: a(t) = -cos(t)i - sin(t)j
2. Integrate a(t) with respect to t to find v(t):
v(t) = ∫(-cos(t)i - sin(t)j) dt = (sin(t)i + cos(t)j) + C, where C is a constant vector.
3. Apply initial condition v(0) = j + k:
v(0) = sin(0)i + cos(0)j + C = j + k
C = -i + j + k
4. The velocity function is: v(t) = sin(t)i + cos(t)j - i + j + k
Now let's find the position vector r(t):
5. Integrate v(t) with respect to t to find r(t):
r(t) = ∫(sin(t)i + cos(t)j - i + j + k) dt = (-cos(t)i + sin(t)j + t(k) + D, where D is another constant vector.
6. Apply initial condition r(0) = i:
r(0) = -cos(0)i + sin(0)j + 0(k) + D = i
D = i
7. The position function is: r(t) = -cos(t)i + sin(t)j + tk + i
Finally, let's find the position at time t = 9:
8. r(9) = -cos(9)i + sin(9)j + 9k + i
This gives you the position vector r(9) as a function of sin(9) and cos(9).
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Given logaMN = 6, log aN/M = 2 and logaN^m = 16, find M.
The value of M is a^4.
Given the information, we can express the given logarithms as follows:
1) log_a(MN) = 6
2) log_a(N/M) = 2
3) log_a(N^m) = 16
From equation (1), we can write:
MN = a^6
From equation (2), we can write:
N/M = a^2 → N = a^2 * M
Now, substitute N from equation (2) into equation (3):
log_a((a^2 * M)^m) = 16
Using the power rule of logarithms, we get:
m * log_a(a^2 * M) = 16
Since log_a(a^2 * M) = 2log_a(a) + log_a(M) = 2 + log_a(M), we have:
m * (2 + log_a(M)) = 16
We don't have enough information to determine the value of 'm', but we don't need it to find the value of 'M'.
Now, substitute N back into the equation MN = a^6:
M * a^2 * M = a^6
Divide both sides by M * a^2:
M = a^(6-2) = a^4
So, the value of M is a^4.
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In a hypothesis test for a mean in one population, where you have H subscript 0 colon space mu space equals space 40 comma space H subscript A colon space mu space not equal to space 40 and the population standard deviation is sigma space equals space 12, what are the critical value(s) of the sample mean x with bar on top if your sample size is 36 and the significance level alpha = 0. 05?
Group of answer choices
Using the t-distribution table with a sample size of 36 and a significance level of 0.05, we find the critical t-value to be ±2.03 (with 34 degrees of freedom, which is n-1).
What are the critical values of the sample mean for a hypothesis test with a sample size of 36, population standard deviation of 12, significance level of 0.05, and null hypothesis of μ = 40?
To explain, we use the t-distribution to find the critical values because the population standard deviation is known. Since the alternative hypothesis is two-tailed (H_A: μ ≠ 40), we need to find two critical values.
With a sample size of 36, the degrees of freedom are 34 (n-1), so we use a t-distribution table with 34 degrees of freedom and a significance level of 0.05. From the table, we find the critical t-value to be ±2.03.
Therefore, if the calculated t-value falls outside of this range, we can reject the null hypothesis H0: μ = 40 in favor of the alternative hypothesis H_A: μ ≠ 40.
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Sally earns a weekly salary of $450 plus a 6. 5% commission on sales at a boutique. How much would she make in a work week if she sold $650 worth of merchandise?
To find out Sally's total earnings for the week, we need to consider her base salary and the commission on her sales. Her base salary is $450, and she earns a 6.5% commission on $650 worth of merchandise.
First, let's calculate her commission:
6.5% of $650 = 0.065 * $650 = $42.25
Now, we can add her base salary to the commission:
Total earnings = Base salary + Commission
Total earnings = $450 + $42.25
Total earnings = $492.25
So, Sally would make $492.25 in a work week if she sold $650 worth of merchandise.
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HELPPPPPPPPPp WILL GIVE BRAINLEISTTT!!!
Answer:
100
Step-by-step explanation:
i think this is right
Among 130 pupils, 30 liked both biscuits and chocolates, 10 liked neither and twice as many as liked biscuits liked chocolates.
I) How pupils liked: chocolates, biscuits and exactly one of the two.
The number of pupils who liked both biscuits and chocolates is 30.
The number of pupils who liked neither biscuits nor chocolates is 10.
Let's assume that the number of pupils who liked only biscuits is x, and the number of pupils who liked only chocolates is y.
According to the problem, twice as many pupils liked chocolates as those who liked biscuits. Mathematically, we can write this as:
y = 2x
Now, let's find the total number of pupils who liked at least one of the two:
Total = P(Biscuits) + P(Chocolates) - P(Biscuits and Chocolates)
Total = x + y + 30
Total = x + 2x + 30
Total = 3x + 30
We know that the total number of pupils is 130, and the number of pupils who liked neither is 10. Therefore,
Total = P(All pupils) - P(Neither)
130 = x + y + 30 + 10
130 = x + y + 40
130 - 40 = x + y
90 = x + y
We can now solve these two equations to get the values of x and y:
3x + 30 = 90
3x = 60
x = 20
y = 2x = 40
Therefore, 20 pupils liked only biscuits, 40 pupils liked only chocolates, and 30 pupils liked both biscuits and chocolates. And, 40 pupils liked exactly one of the two.
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Determine where the absolute extrema of f(x)= 4x/ x²+1 on the interval [-4,0] occur. 1. The absolute maximum occurs at x= 2. The absolute minimum occurs at x =
The absolute maximum of f(x) = 4x / (x² + 1) on the interval [-4,0] occurs at x = 2 and the absolute minimum occurs at x = -4.
To find the absolute extrema, we first find the critical points by setting the derivative of f(x) equal to zero:
f'(x) = (4(x² + 1) - 8x²) / (x² + 1)² = 0
Simplifying, we get:
4 - 4x² = 0
x² = 1
x = ±1
Since x = -4 and x = 0 are also endpoints of the interval, we evaluate f(x) at these five points:
f(-4) = -8/17
f(-1) = -4/5
f(0) = 0
f(1) = 4/5
f(2) = 8/5
Thus, the absolute maximum occurs at x = 2, where f(x) = 8/5, and the absolute minimum occurs at x = -4, where f(x) = -8/17.
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