Twice the difference of a number and 4 to is 5

The absolute maximum value of f(x) = x³ - 6x² + 9x + 6 on [-1, 6] is 84, which occurs at x = 6, and the absolute minimum value is 0, which occurs at x = 3.

To find the absolute maximum and minimum values of the function f(x) = x³ - 6x² + 9x + 6 on the interval [-1, 6], we need to find the critical points of the function and evaluate the function at the endpoints of the interval.

First, we find the derivative of the function

f'(x) = 3x² - 12x + 9

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

3x² - 12x + 9 = 0

Dividing both sides by 3, we get

x² - 4x + 3 = 0

Factoring, we get

(x - 1)(x - 3) = 0

So the critical points are x = 1 and x = 3.

Next, we evaluate the function at the endpoints of the interval

f(-1) = (-1)³ - 6(-1)² + 9(-1) + 6 = 2

f(6) = 6³ - 6(6)² + 9(6) + 6 = 84

Now we need to evaluate the function at the critical points

f(1) = 1³ - 6(1)² + 9(1) + 6 = 10

f(3) = 3³ - 6(3)² + 9(3) + 6 = 0

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The given question is incomplete, the complete question is:

Find the absolute maximum and absolute minimum values off on the given interval. (If an answer does not exist, enter DNE.) f(x) = x³ - 6x² + 9x + 6 on [-1, 6]

Based on the information, the data table can be represented as follows:
```

Wins                 No Clear  Winner   Totals
Initiator Loses     18           14             32
Fight                    24          75            99
No Fight              15           104           119
Totals                  57           193          250
```

Here's a breakdown of the data:
1. Initiator Loses:
  - 18 wins
  - 14 no clear winner
  - 32 total outcomes
2. Fight:
  - 24 wins
  - 75 no clear winner
  - 99 total outcomes
3. No Fight:
  - 15 wins
  - 104 no clear winner
  - 119 total outcomes
4. Totals:
  - 57 total wins
  - 193 total no clear winner
  - 250 total outcomes
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Approximately 84% (34% + 50%) of the jumpers were in the group jumping below 79.5 inches. This can be answered by the concept of Standard deviation.

In a high school high jump contest, the mean height was 76 inches, and the standard deviation was 3.5 inches. To find the percentage of jumpers below 79.5 inches, we first need to determine how many standard deviations away 79.5 inches is from the mean.

To do this, subtract the mean from 79.5 inches and divide by the standard deviation:
(79.5 - 76) / 3.5 = 3.5 / 3.5 = 1

So, 79.5 inches is 1 standard deviation above the mean. According to the empirical rule, approximately 68% of the data falls within 1 standard deviation of the mean in a normal distribution. Since we are looking for jumpers below 79.5 inches, we need to consider the lower half of this 68%, which is 34%. Additionally, 50% of the data is below the mean.

Therefore, approximately 84% (34% + 50%) of the jumpers were in the group jumping below 79.5 inches.

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To find the critical value at 0.05, we need to use the chi-square distribution. Since the null hypothesis (H0) asserts that the variance is less than 6, we can use a one-tailed test with alpha = 0.05.

To calculate the critical value, we need to first find the degrees of freedom (df) which is equal to n-1, where n is the sample size.

In this case, df = 26-1 = 25.

Next, we need to find the chi-square value for a one-tailed test with 25 degrees of freedom and alpha = 0.05.

We can use a chi-square distribution table or a calculator to find this value. Using a calculator, we get: χ² = CHISQ.INV(0.05, 25) = 37.65248

Therefore, the critical value for this test is 37.65248. Any calculated chi-square value greater than this critical value would lead to rejection of the null hypothesis.

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That is the solution to the differential equation ds/dt = -4 + 3cos(t) with the initial condition s(0) = 2.

To solve this problem, we need to integrate both sides of the differential equation with respect to t and then use the initial condition to find the constant of integration. Here are the steps:

Integrating both sides with respect to t, we get:

∫ds = ∫(-4 + 3cos(t)) dt

The integral on the left side is simply s(t), so we have:

s(t) = -4t + 3sin(t) + C

where C is the constant of integration.

Now we can use the initial condition s(0) = 2 to find the value of C:

s(0) = -4(0) + 3sin(0) + C = 0 + 0 + C = C

Therefore, C = 2, and the function s(t) is:

s(t) = -4t + 3sin(t) + 2

That is the solution to the differential equation ds/dt = -4 + 3cos(t) with the initial condition s(0) = 2.

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9a) The function form is f(x) = -4(-3)^(x-1). The sequence is not an exponential function because the base is negative.

9b) The function form is g(x) = -16 * (-1/4)^(x-1). The sequence is not an exponential function because the base is between 0 and 1.

10) Geometric sequences and exponential functions are closely related, but not all geometric sequences are exponential functions. Every geometric sequence with a positive base can be represented as an exponential function with the same base.

9)

a) The explicit formula for sequence E is y = -4 (-3)^(x-1). To write it in function form, we can define a function f(x) = -4(-3)^(x-1), where f(x) represents the value of the sequence at the xth term.

The reason why this geometric sequence is not an exponential function is that the base (-3) is negative. Exponential functions have positive bases, whereas geometric sequences can have either positive or negative bases.

b) The explicit formula for sequence F is y = -16 * (-1/4)^(x-1). To write it in function form, we can define a function g(x) = -16 * (-1/4)^(x-1), where g(x) represents the value of the sequence at the xth term.

Similar to sequence E, the reason why this geometric sequence is not an exponential function is that the base (-1/4) is between 0 and 1, whereas exponential functions have bases greater than 1 or between 0 and 1.

10) Geometric sequences and exponential functions are closely related. In fact, every geometric sequence with a positive base can be represented as an exponential function with the same base.

For example, the geometric sequence with a constant ratio of 2 can be written as the exponential function f(x) = 2^x. Similarly, a geometric sequence with a constant ratio of 1/3 can be written as the exponential function g(x) = (1/3)^x.

However, as we saw in the previous question, geometric sequences with negative or fractional bases are not exponential functions. Therefore, not all geometric sequences are exponential functions.

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Here are the ages of the generations in 2022:

Baby Boomer Generation: Born between 1946 and 1964, so in 2022, they will be between 58 and 76 years old.

Generation X: Born between 1965 and 1980, so in 2022, they will be between 42 and 57 years old.

Millennials or Generation Y: Born between 1981 and 1996, so in 2022, they will be between 26 and 41 years old.

Generation Z: Born between 1997 and 2012, so in 2022, they will be between 10 and 25 years old.

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By determining how many basketball players met the cutoff jump height and how many did not meet the cutoff height for a countermovement jump test using the My Jump and VertJump mobile apps, we would be examining the players' ability to generate power and explosiveness in their vertical jump.

This is a critical skill for basketball players as it allows them to jump higher for rebounds, block shots, and score points. Knowing who met the cutoff height would give coaches and trainers an idea of which players possess this crucial skill and can be relied upon to perform well in the game. On the other hand, those who did not meet the cutoff height may need to work on their jumping ability to improve their performance on the court.

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The value of the integral is: S√2 1 (u⁷/2 - 1/u⁵)du = 28/9 - 15/4

= (112/36) - (135/36)

= -23/36.

To evaluate the integral S√2 1 (u⁷/2 - 1/u⁵)du, we can use the linearity property of integration and split the integrand into two separate integrals:

S√2 1 (u⁷/2 - 1/u⁵)du = S√2 1 u⁷/2 du - S√2 1 1/u⁵ du

Now, we can integrate each of these separate integrals:

S√2 1 u⁷/2 du = (2/9) u⁹/2 |1 √2 = (2/9) * (2√2⁹/2 - 1)

= (4/9) (√2⁴ - 1)

= (4/9) (8 - 1)

= 28/9

S√2 1 1/u⁵ du = (-1/4) u⁻⁴ |1 √2 = (-1/4) * (1 - 2⁴)

= (-1/4) * (-15)

= 15/4

Therefore, the value of the integral is: S√2 1 (u⁷/2 - 1/u⁵)du = 28/9 - 15/4

= (112/36) - (135/36)

= -23/36.

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Answer:

The answer to your problem is:

Part A. Box B:

H,B | H,R | H, Y

T,B | T,R | T, U

Part B. 0.2

Step-by-step explanation:

Part A.

We can put it to this diagram shown:

[tex]\left[\begin{array}{ccc} &H& \\l&l&l\\B&R&Y\end{array}\right][/tex] “ l “ Representing what H is going to. ( Same with second )

[tex]\left[\begin{array}{ccc} &T& \\l&l&l\\B&R&Y\end{array}\right][/tex] “ l “ Represening what T is going to

So if we complete it, it will equal:

H,B | H,R | H, Y

T,B | T,R | T, U

Part B.

We will just solve:

[tex]\frac{1}{2} * \frac{2}{1+2+2} = \frac{1}{2} * \frac{2}{J} = \frac{1}{J} = 0.2[/tex]

0.2 being our answer

Thus the answer to your problem is:

Part A. Box B:

H,B | H,R | H, Y

T,B | T,R | T, U

Part B. 0.2

Answer:

The answer for

a)35

b)35

Step-by-step explanation:

a)34.9961----->2d.p

=35=to 2.d.p

b)34.9961------>the nearest tenth

=35

Answer:

Step-by-step explanation:

Find your fractional portion and multiply by the area

=[tex]\frac{45}{360}[/tex] * [tex]\pi[/tex] r²      substitute r=10 and simplify

=[tex]\frac{45*10}{360}[/tex] [tex]\pi[/tex]         Reduce the fraction

=[tex]\frac{5\pi }{4}[/tex]       D

The correlation will not be −0.44 based solely on the slope of the regression line. (option c).

Let X and Y be the vectors of standardized values of X and Y, respectively, for all the subjects. Then, the least-squares regression line can be written as:

Y = βX

where β is the slope of the regression line. To find the intercept, we need to solve for the value of Y when X = 0:

Y = β(0) = 0

This means that the intercept of the regression line in the standardized coordinate system is 0. To find the intercept in the original coordinate system, we need to transform this point back using the formula for standardization:

Y = σY(Y) + μY

where σY is the standard deviation of Y and μY is the mean of Y. Since y = 0, we have:

Y = σY(0) + μY = μY

So, the intercept of the regression line in the original coordinate system is equal to the mean of Y. Therefore, we cannot conclude that the intercept will be −0.44 or 1.0.

Hence the correct option is (c).

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Complete Question

When we standardize the values of a variable, the distribution of standardized values has mean 0 and standard deviation 1. Suppose we measure two variables X and Y on each of several subjects. We standardize both variables and then compute the least-squares regression line. Suppose the slope of the least-squares regression line is 20.44. We may conclude that

a. The intercept will also be −0.44.

b. The intercept will be 1.0.

c. The correlation will not be 1/−0.44.

Answer:

B. The product will be less than 16.

To solve the expression, we can multiply 16 by 2/3:

16 x 2/3 = (16 x 2) / 3 = 32 / 3

This fraction is between 10 and 11, which means the product is less than 16.

The result after the evaluation of definite integral is -12 + 4π, under the given condition that  [tex]I = \int\limits^0_4 (2+\sqrt{16} -x^{2})dx[/tex]needs to be  interpreted concerning the known areas.

The given definite integral  [tex]I = \int\limits^0_4 (2+\sqrt{16} -x^{2})dx[/tex]could be placed as the difference between two areas
The area under the curve of the function (2+√16-x²) from x=0 to x=-4
The area of a rectangle with base 4 and height 2.

The area under the curve can be evaluated by finding the area of a quarter circle with radius 4 and subtracting it from the area of a triangle with base 4 and height 2.
The quarter circle has an area of πr²/4 = π(4)²/4 = 4π
The triangle has an area of (1/2)(4)(2) = 4.
Therefore, the area under the curve is 4π - 4.
The area of a rectangle with base 4 and height 2 is simply 8.

Now,
[tex]I = \int\limits^0_4 (2+\sqrt{16} -x^{2})dx[/tex]
= (4π - 4) - 8
= -12 + 4π
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This series can be used to approximate f(x) for values of x close to π/6.

The Taylor series of f(x) = sin(3x) centered at x = π/6 is given by the formula:

f(x) = ∑(n=0 to infinity) [(-1)ⁿ * 3²ⁿ⁺¹/ (2n+1)!] * (x - π/6)²ⁿ⁺¹

To find the coefficients of this series, we need to evaluate f and its derivatives at x = π/6.

f(π/6) = sin(3π/6) = sin(π/2) = 1

f'(π/6) = 3cos(3π/6) = 0

f"(π/6) = -9sin(3π/6) = -9

f"'(π/6) = -27cos(3π/6) = 27

f(4)(π/6) = 81sin(3π/6) = 0

Using these values, we can plug them into the Taylor series formula and simplify to get:

f(x) = 1 - 9/2(x - π/6)² + 27/4(x - π/6)³ - 81/40(x - π/6)⁵ + ...

In other words, the Taylor series of f(x) = sin(3x) centered at x = π/6 is a power series with coefficients that depend on the derivatives of f at π/6.

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a)The only value of p for which lim f(x) exists is p = 1.b) The limit of f(x) doesnt exist.

(a)We have given the equation f(x) = 24 – 24 cos(2x) – 48x^2. To find the value of the constant p for which lim f(x) exists, we need to simplify f(x) and check the left and right-hand limits as x approaches 0.

f(x) = 24 – 24 cos(2x) – 48[tex]x^{2}[/tex]

= 24 (1 – cos(2x)) – 48[tex]x^{2}[/tex]

= 48 [tex]sin^{2} x^{}[/tex] – 48[tex]x^{2}[/tex]

Now, as x approaches 0, sin(x) ~ x. So, we can replace [tex]sin^{2} x^{}[/tex] with [tex]x^{2}[/tex] in the above expression.

f(x) = 48 [tex]sin^{2} x[/tex] – 48[tex]x^{2}[/tex]

= 48[tex]x^{2}[/tex] – 48[tex]x^{2}[/tex] = 0

Therefore, the only value of p for which lim f(x) exists is p = 1.

(b) Using p = 1, we have:

lim f(x) = lim [6x sin(6x) + px] / [[tex]x^{3}[/tex]]

= lim [6 sin(6x) + p/[tex]x^{2}[/tex]] / 3[tex]x^{2}[/tex] (Dividing numerator and denominator by [tex]x^{2}[/tex])

= 6 lim sin(6x)/6x + p/3 lim 1/[tex]x^{2}[/tex] (Applying limit rules)

Now, lim sin(6x)/6x = 1 (using the limit definition of derivative)

And lim 1/[tex]x^{2}[/tex] = infinity (as x approaches 0 from both sides)

Therefore, lim f(x) = 6 + infinity = infinity, limit doesn't exist.

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Answer:

Step-by-step explanation:

You want the measures of the angles marked m° and n° in the given figure.

The Claim, Evidence, Reasoning (CER) model tells us an explanation consists of:

Claim: The measure of m° is 63°.

Evidence: Angle m° is one of three angles in the figure that form a straight angle.

Reasoning: The measure of a straight angle is 180° (definition). The sum of the angles is equal to the whole (angle addition theorem).

  m° +90° +27° = 180°

  m° = 63° . . . . add -117° to both sides (addition property of equality)

Claim: The measure of n° is 28°.

Evidence: Angle n° is one of two angles in the figure that form a right angle.

Reasoning: The square corner signifies a right angle, whose measure is 90°. The angle addition theorem tells us that angle is the sum of the two angles into which it is divided:

  90° = 62° + n°

  28° = n° . . . . . . . add -62° to both sides (addition property of equality)

The boundary value of F using an F-distribution calculator is 6.2561. So, the correct option is option c. 6.2561.

To find the boundary value of F where P(FDATA < F) = 0.95 for an F distribution with df1 = 4 and df2 = 5 if FDATA follows an F distribution, follow the steps given below:

1. Identify the degrees of freedom: df1 = 4 and df2 = 5.
2. Determine the desired probability: P(FDATA < F) = 0.95.
3. Consult an F-distribution table or use an online calculator or statistical software to find the F-value corresponding to the given degrees of freedom and probability.

So, using an F-distribution calculator, the boundary value of F where P(FDATA < F) = 0.95 is approximately 6.2561. Therefore, the correct answer is 6.2561.

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a) To maximize profit, the monopoly should produce approximately 2.29 hundred units of the product

b) The maximum profit it can earn is $4085.61.

a) To find the quantity that will give maximum profit, we need to maximize the profit function, which is given by:

P(x) = (1836 −1/3(x)²)x − (900 + 10x + x²)x

Simplifying this expression, we get:

P(x) = 1836x − 1/3x³ − 900x − 10x² − x³

P(x) = -4/3x³ - 10x² + 936x

To find the maximum profit, we need to find the critical points of this function. Taking the derivative of P(x) with respect to x and setting it equal to zero, we get:

P'(x) = -4x² - 20x + 936 = 0

Solving for x, we get:

x = 22.87 or x = -10.26

Since production is limited to 1000 units, we can discard the negative value. Therefore, the quantity that will give maximum profit is approximately 2.29 hundred units.

b) To find the maximum profit, we can substitute this value of x into the profit function:

P(2.29) = (1836 −1/3(2.29)²)(2.29) − (900 + 10(2.29) + (2.29)²)(2.29)

P(2.29) = 4085.61

Therefore, the maximum profit is $4085.61.

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For $23, you can rent a computer at the local copy center for approximately 25 minutes and 27 seconds.

A declarative sentence that can be either true or false, but not both, is called a statement.

Let's call the amount of time that can be rented "t" (in minutes).

We know that there is a $9 setup charge and an additional $5.50 for every 10 minutes, so the total cost C (in dollars) can be expressed as:

C = 9 + 5.5 × (t / 10)

We want to find out how much time can be rented for $23, so we can set C equal to 23 and solve for t:

23 = 9 + 5.5 × (t / 10)

Subtracting 9 from both sides, we get:

14 = 5.5 × (t / 10)

Multiplying both sides by 10/5.5, we get:

t = 25.45 minutes

So, for $23, you can rent a computer at the local copy center for approximately 25 minutes and 27 seconds.

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The statement "a manager has only 200 tons of plastic for his company" is an example of a constraint.

A constraint is a limitation or restriction that affects the decision-making process.

In this case, the amount of plastic available to the manager is a constraint that will influence his or her decisions about how to allocate resources and manage the company's operations.

Constraints are an important consideration in many decision-making contexts as they can significantly affect the feasibility and effectiveness of different options.

For example,

A company that is constrained by limited financial resources may need to prioritize investments and expenses in order to achieve its goals.

In contrast to constraints, objectives are the specific goals or outcomes that a manager aims to achieve through his or her decisions and actions.

Parameters, on the other hand, refer to the specific values or variables that are used to define a particular situation or problem.

Decisions, meanwhile, are the choices that a manager makes in response to a given situation or problem.

In this case, the manager may need to make decisions about how to best use the limited amount of plastic available to the company, taking into account factors such as production goals, quality standards, and financial considerations.

Overall, the constraint of limited plastic availability is an important consideration that will impact the manager's decisions and actions, and must be taken into account in the overall decision-making process.

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The final expression for f(t) is f(t) = (1/6)t⁶ - (1/6)t⁻⁶ + 1/6

To find f(t), we need to integrate f'(t) with respect to t. Since the derivative of f(t) involves two terms, we need to split the integral into two parts:

∫[t⁵ + 1/t⁷] dt = ∫t⁵ dt + ∫1/t⁷ dt

Integrating the first part gives:

∫t⁵ dt = (1/6)t⁶ + C₁

where C₁ is the constant of integration.

Integrating the second part gives

∫1/t⁷ dt = (-1/6)t⁻⁶ + C₂

where C₂ is the constant of integration.

Therefore, we have

f(t) = (1/6)t⁶ - (1/6)t⁻⁶ + C

where C = C₁ + C₂ is the constant of integration

To find the value of C, we use the initial condition f(1) = 7

f(1) = (1/6)(1)⁶ - (1/6)(1)⁻⁶ + C = 7

Simplifying this expression gives:

C = 7 + (1/6) - (1/6)(1)⁻⁶ = 7 + 1/6 - 1 = 7 + 1/6 - 6/6

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The given question is incomplete, the complete question is:

Find f. f'(t) = t⁵ + 1/t⁷ t> 0, f(1) = 7. f(t) =

The number of cups of pretzels that Carter will need is 5 ¹ / ₃ cups .

The original formula of the ratio between pretzels and raisins, would be:

1 1 / 3 : 1 / 4

4 / 3 : 1 / 4

Seeing as Carter wants to use 1 full cup of raisins, this means that the ratio will have to be increased by 4 on both sides. This would make the raisins, one cup. And would make the pretzels:

= 4 / 3  x 4

= 16 / 3

= 5 ¹ / ₃ cups

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The linearization l(x) of the function at a is -(x-π/2).

We have, f(x) = cos (x), x =π/22

Now, differentiating on both sides

f'(x) = -sin (x)

At x=π/2

y = f(π/2) = cos (π/2) = cos (90°)= 0

and f'(π/2) = -sin(π/2) = -sin (90°) = -1

Now, The linearization is the tangent line

L(x)= f(a) + f'(a)(x-a)

     = 0 + (-1)(x-π/2)

Therefore, the linearization l(x) of the function at a is -(x-π/2)

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Both sides of the equation are equal.

Given the identity:

5 cos² β - 5 sin² β = 10 cos² β - 5

We can use the Pythagorean identity, which states that:

sin² β + cos² β = 1

Now, we can rewrite the given equation by expressing sin² β in terms of cos² β:

5 cos² β - 5 (1 - cos² β) = 10 cos² β - 5

Next, distribute the -5:

5 cos² β - 5 + 5 cos² β = 10 cos² β - 5

Combine like terms:

10 cos² β - 5 = 10 cos² β - 5

The identity is now verified. Both sides of the equation are equal.

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The general solution to the given differential equation is:

If [tex]b^2[/tex] - 288 > 0: [tex]y(t) = c1e^{(b + \sqrt{(b^2 - 288)} )t/16} + c2e^{(b - \sqrt{(b^2 - 288)} )t/16} - 1/3[/tex]

If [tex]b^2[/tex] - 288 = 0:[tex]y(t) = (c1 + c2t)e^{bt/16 } - 1/3[/tex]

If[tex]b^2[/tex] - 288 < 0: [tex]y(t) = e^{bt/16} (c1cos[wt/16] + c2sin[wt/16]) - 1/3, \\where w = \sqrt{(288 - b^2)/16.}[/tex]

To find the general solution to the given differential equation:

8y'' - by' + 9y = -8

We first need to find the roots of the characteristic equation:

[tex]8m^2 - bm + 9 = 0[/tex]

Using the quadratic formula:

[tex]m = [b +/- \sqrt{(b^2 - 4(8)(9))]/(2(8))}][/tex]

[tex]m = [b +/- \sqrt{(b^2 - 288)]/16} ][/tex]

The roots of the characteristic equation are:

[tex]m1 = [b + \sqrt{(b^2 - 288)]/16} ][/tex]

[tex]m2 = [b - \sqrt{(b^2 - 288)]/16}][/tex]

Depending on the value of b, there are three possible cases:

Case 1: [tex]b^2[/tex]- 288 > 0, which implies that there are two distinct real roots.

In this case, the general solution is:

[tex]y(t) = c1e^{m1t} + c2e^{m2t} - 1/3[/tex]

where c1 and c2 are constants determined by the initial conditions.

Case 2: [tex]b^2[/tex] - 288 = 0, which implies that there is one repeated real root.

In this case, the general solution is:

[tex]y(t) = (c1 + c2t)e^{mt} - 1/3[/tex]

where c1 and c2 are constants determined by the initial conditions.

Case 3: [tex]b^2[/tex] - 288 < 0, which implies that there are two complex conjugate roots.

In this case, the general solution is:

[tex]y(t) = e^{bt/16}(c1cos(wt/16) + c2sin(wt/16)) - 1/3[/tex]

where c1 and c2 are constants determined by the initial conditions, and [tex]w = \sqrt{(288 - b^2)/16.}[/tex]

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The joint distribution of {X_t} and {X_{t+k}} is the same as the joint distribution of {X_{t+n}} and {X_{t+n+k}}, and hence {X_t} is strictly stationary.

The time series {X_t, t=0, ±1, ±2, ...} is said to be strictly stationary if its joint distribution is invariant under time shifts.

i.e., For any set of integers n and k and any permutation of their sum, the joint distribution of {X_t} and {X_t+k} is the same as the joint distribution of {X_t+n} and {X_t+n+k}, respectively.

(a) x_t = 4sin(ωt) where ω is a fixed frequency and {e_t} is a sequence of independent and identically distributed (i.i.d.) N(0,σ^2) random variables.

To check if {X_t} is strictly stationary, we need to check if its joint distribution is invariant under time shifts.

Let n and k be any integers and consider the joint distribution of {X_t} and {X_{t+k}}.

We have:

E[X_t] = E[4sin(ωt)] = 0 (since sin(ωt) is an odd function and we are integrating over a full period)

Cov(X_t, X_{t+k}) = Cov(4sin(ωt), 4sin(ω(t+k)))

= 16Cov(sin(ωt), sin(ω(t+k)))

= 8Cov(cos(ωt-k), sin(ωt))

= 0 (since cos(ωt-k) and sin(ωt) are orthogonal)

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The derivatives of y in each case is:

(a) [tex]dy/dx = u(dv/dx) + v(du/dx) = sin(x) * (-sin(x)) + cos(x) * cos(x) = -sin^2(x) + cos^2(x).[/tex]

(b) [tex]dy/dx = u(dv/dx) + v(du/dx) = x * (cos(x^3) * 3x^2) + sin(x^3) * 1 = 3x^3*cos(x^3) + sin(x^3).[/tex]

(a) y = sin(x)
To find the derivative of y with respect to x, use the chain rule. The derivative of sin(x) with respect to x is cos(x).
So, dy/dx = cos(x).

(b) y = sin(x) cos(x)
To find the derivative, use the product rule. Let u = sin(x) and v = cos(x).
The derivative of u with respect to x is du/dx = cos(x), and the derivative of v with respect to x is dv/dx = -sin(x).

Apply the product rule: [tex]dy/dx = u(dv/dx) + v(du/dx) = sin(x) * (-sin(x)) + cos(x) * cos(x) = -sin^2(x) + cos^2(x).[/tex]

(c) y = x * sin(x^3)
Here, use the product rule again. Let u = x and v = sin(x^3).
The derivative of u with respect to x is du/dx = 1, and the derivative of v with respect to x requires the chain rule.

The outer function is sin(w) and the inner function is[tex]w = x^3. So, dw/dx = 3x^2 and dv/dw = cos(w).[/tex]

By the chain rule, [tex]dv/dx = dv/dw * dw/dx = cos(x^3) * 3x^2.[/tex]

Now, apply the product rule: [tex]dy/dx = u(dv/dx) + v(du/dx) = x * (cos(x^3) * 3x^2) + sin(x^3) * 1 = 3x^3*cos(x^3) + sin(x^3).[/tex]

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Answer:

E, A, C!

Step-by-step explanation:

Have a good day

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