Let f be the function defined by f(x)=xlnx for x>0. On what open interval is f decreasing?

Central Limit Theorem forms the foundation for the inferential statistics branch of statistics

The Central Limit Theorem is a statistical concept that describes the behavior of the mean of a sufficiently large sample size drawn from any population with a finite mean and variance.

The Central Limit Theorem is a fundamental concept in statistics that states that, under certain conditions, the distribution of sample means approaches a normal distribution regardless of the shape of the population distribution. This theorem provides a basis for inferential statistics, which involves making inferences about a population based on a sample.

It allows us to use sample statistics to estimate population parameters with a certain level of confidence. This theorem has practical applications in many areas of research, such as in quality control, social sciences, and healthcare.

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BMED is a trapezoid and OD is equals to BD/2. Proof of these is given below.

It should be noted that to prove that BMED is a trapezoid, we need to show that either BM || ED or BM || DE. We will show that BM || DE.

Since O is the midpoint of AM, we have OA = OM, and thus triangle ODM is isosceles. Therefore, OD = DM.

Also, since E is the midpoint of DC, we have DE || AB (because AB is also a mid-segment of triangle ADC). Therefore, angle ADE = angle ABC.

Using the fact that angles in a triangle add up to 180 degrees, we have:

angle EDC + angle ADE + angle CED = 180 degrees

Substituting angle ADE = angle ABC, we get:

angle EDC + angle ABC + angle CED = 180 degrees

But angle CED = angle BCD (because they are alternate interior angles formed by transversal BD cutting parallel lines DC and AB). Therefore:

angle EDC + angle ABC + angle BCD = 180 degrees

This means that angles BCD, ABC, and EDC are all on the same line, and thus angle BCD + angle ABC = 180 degrees.

Now, using the fact that angles in a triangle add up to 180 degrees, we have:

angle BAC + angle ABC + angle ACB = 180 degrees

Substituting angle BCD + angle ABC = 180 degrees, we get:

angle BAC + angle BCD + angle ACB = 180 degrees

Since triangle BCD is isosceles (because BD is an angle bisector), we have angle BCD = angle CBD. Therefore:

angle BAC + angle CBD + angle ACB = 180 degrees

But angle ACD = angle ACB + angle CBD, so:

angle BAC + angle ACD = 180 degrees

This means that triangle ABC is similar to triangle ACD.

Now, since BM is a median of triangle ABC, we have:

BM = 1/2 AC

But triangle ADC is similar to triangle ABC, so:

AC = 2 AD

Therefore:

BM = AC/2 = AD

So we have BM || DE, and BM = DE, which means that BMED is a trapezoid.

2. To prove that D is the midpoint of AE, we will use the fact that BMED is a trapezoid. Since BM || DE, we have:

BD/DM = BE/EM

But DM = OD (since triangle ODM is isosceles) and EM = DC/2 = AC/4 (since E is the midpoint of DC and AC is a mid-segment of triangle ADC).

Therefore:

BD/OD = BE/(AC/4)

But we also have:

BE = BD + DE = BD + BM = 2BD

Substituting this into the previous equation, we get:

BD/OD = 2BD/(AC/4)

Simplifying:

BD/OD = 8BD/AC

This means that:

OD = AC/8

But we also know that AC = 2AD, so:

OD = AD/4

Since O is the midpoint of AM, we also have:

OD = OM = AM/2

Therefore:

AD/4 = AM/2

This means that:

AD = 2AM/4 = AM/2 = OD

So D is the midpoint of AE, and CD = 2AD.

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In a triangle ABC. [AM] is the median relative to [BC] and O is the midpoint of [AM]. (BO) cuts [AC] at D. Let E be the midpoint of [DC]. 1) Prove that BMED is a trapezoid. 2) Prove that D is the midpoint of [AE); deduce that CD = 2AD. 3) Prove that OD == BD.

Answer:

D, A, B, C

Step-by-step explanation:

-12/3, or -4, is lowest

-11/3, or -3.6, is 0.3 more, therefore closer to 0.

-2/3, or -0.6, is much closer to 0, precisely 3 units closer.

1/3, or 0.3, is the only positive number, so it is the most amount.

2.29 is less than the MAD of 2.53, it would not be unlikely to get a box with 31 red pencils based on the MAD for the given data set.

To determine whether it would be unlikely to get a box with 31 red pencils based on the Mean Absolute Deviation (MAD) for the given data set, we first need to calculate the MAD.

Step 1: Calculate the mean of the data set

The given data set is: 22, 28, 30, 28, 35, 30, 28

The mean is calculated by adding up all the numbers and dividing by the total number of values:

Mean = (22 + 28 + 30 + 28 + 35 + 30 + 28) / 7 = 201 / 7 = 28.71 (rounded to two decimal places)

Step 2: Calculate the absolute deviations

The absolute deviation for each data point is calculated by finding the absolute difference between the data point and the mean:

|22 - 28.71| = 6.71

|28 - 28.71| = 0.71

|30 - 28.71| = 1.29

|28 - 28.71| = 0.71

|35 - 28.71| = 6.29

|30 - 28.71| = 1.29

|28 - 28.71| = 0.71

Step 3: Calculate the MAD

The MAD is the mean of the absolute deviations:

MAD = (6.71 + 0.71 + 1.29 + 0.71 + 6.29 + 1.29 + 0.71) / 7 = 17.71 / 7 = 2.53 (rounded to two decimal places)

Now, to determine whether it would be unlikely to get a box with 31 red pencils, we can compare 31 with the MAD. If the difference between 31 and the mean (28.71) is greater than the MAD (2.53), then it would be unlikely.

Difference = 31 - 28.71 = 2.29

Since 2.29 is less than the MAD of 2.53, it would not be unlikely to get a box with 31 red pencils based on the MAD for the given data set.

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Answer:

There are different methods to find the horizontal asymptote of a function, but one common approach is to take the limit of the function as x approaches positive or negative infinity. If the limit exists and is a finite number, then that is the horizontal asymptote.

To apply this method to the given function:

y = (-4x^6+6x +3)/(8x^2+9x+3)

As x becomes very large (either positive or negative), the highest power of x in the numerator and denominator dominates the other terms. This is because exponential functions grow or shrink much faster than polynomials as x goes to infinity or negative infinity.

Therefore, we can simplify the function by dividing both numerator and denominator by x^6:

y = (-4 + 6/x^5 + 3/x^6)/(8/x^4 + 9/x^5 + 3/x^6)

As x approaches infinity, all the terms with negative powers of x go to zero, leaving:

y ≈ (-4 + 0 + 0)/(0 + 0 + 0)

This is an indeterminate form of 0/0, which suggests that we need to do more algebraic simplification. One way to do this is to factor out the highest power of x in both numerator and denominator:

y ≈ (-4/x^6 + 6/x^11)/(8/x^4 + 9/x^5 + 3/x^6)

y ≈ (-4/x^6)(1 - 6x^5)/(8/x^4)(1 + 9x/x^4 + 3/x^2)

Now, as x approaches infinity, the fraction becomes dominated by the leading terms:

y ≈ (-4/x^6)(-6x^5)/(8/x^4)(9x/x^4)

y ≈ (24/72)(1/x)

As x goes to infinity, 1/x goes to zero, so:

y ≈ 1/3x

Therefore, the horizontal asymptote of the graph of y = (-4x^6+6x +3)/(8x^2+9x+3) is y = 1/3x.

Here the  p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is sufficient evidence at the 0.05 level that the valve performs above specifications.

We have to evaluate null and alternative hypotheses for the scenario .

In this scenario, the null hypothesis is

H0: μ = 5.8

here

μ = true mean pressure produced by the valve.

Now for the alternative hypothesis is

HA: μ > 5.8

here

μ = true mean pressure produced by the valve.

Therefore to  determine if there is sufficient observation at the 0.05 level that the valve performs above specifications,

Here we can utilize a one-tailed z-test with

α = 0.05

Then,  test statistic can be evaluated as

z = (X' - μ) / (σ / √n)

where:

X' = sample mean

μ = hypothesized value concerning the population mean

σ = population standard deviation

n = sample size

Substituting in our values, we get:

z = (5.9 - 5.8) / (1 / √130) = 1.8856



Now utilizing a z-table  we can evaluate that the p-value associated with this test statistic is approximately 0.0292.

Since this p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is sufficient evidence at the 0.05 level that the valve performs above specifications.

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The time it will take to fill the pool at the given rate would be = 8 mins.

To calculate the volume of the cylindrical swimming pool the formula for the volume of the cylinder should be used which is = πr²h

where r = 5ft

h = 8ft

volume = π×5×5×8 = 200πft³

If 25πft³ = 1 minute

200πft³ = X

Make X the subject of formula;

X = 200/25 = 8mins

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The graph of the data from the table is added as an attachment

From the question, we have the following parameters that can be used in our computation:

Jerome is a photographer. He earns $125 per hour

This means that the equation of the function is

f(x) = 125x

Where x is the number of hours he works

Using the above as a guide, we have the following table of values

x    f(x)

1     125

2    250

3    375

4    500

5    625

6     750

Next, we plot the graph from the table of values and the function f(x) = 125x

The graph of the function is added as an attachment

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A study has a 2 Ã 2 Ã 3 within-groups factorial design. This example has a total of _12_ cell(s). Researchers would need to investigate _three__ main effect(s), _Three two-way interactions __ two-way interaction(s), and _ One three-way interaction__ three-way interaction(s) in this study.

a 2 x 2 x 3 within-groups factorial design, you have:
A total of 2 x 2 x 3 = 12 cells
Three main effects to investigate (one for each factor)
Three two-way interactions to investigate (AxB, AxC, and BxC)
4. One three-way interaction to investigate (AxBxC)
So, in this study, there are 12 cells, researchers would need to investigate 3 main effects, 3 two-way interactions, and 1 three-way interaction.

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Answer:

---------------------

Perimeter is the sum of side lengths.

Given sides:

Find the perimeter by adding up side lengths:

Answer:

C. -1.7

Step-by-step explanation:

-4.2 - x = -2 2/4

-2 2/4 = -10/4 = -2.5

-4.2 - x = -2.5

-x = 1.7

x = -1.7

So, the answer is C. -1.7

Answer:162.1

Step-by-step explanation:We know that the sum of supplementary angles is 180°Let the unknown angle be xThus,100° + x = 180°X = 180° – 100°X = 80°Hence, the unknown angle is 80 …I did it my own way ok…

Answer:

Prove the following identitics: tan 0 + cot 0 = I/(sin 0 cos 0) 0 + tan 0 sec 0 = cosec 0 'sec? 0. 18 cosec = cosec? 0 = tan? 0 cot2 19 sec? 0 sin ' 0 =cos? 0 'sin? 20 cos4 (sec 0 + tan 0) (sec 0 _ tan 0) = 1 2 sin? 0 = cOs? 0 _ 'sin? 0. 22 2 cos? 0_I=1 23 sec? 0 + cosec 0 = sec? 0 cosec? sin? , COs? 24 sec" 0 cosec" 0 = cos 0 sin" 0 25 =I tan 0 + 1 cot? 0 +4 26 (sec? 0 1) (cosec? 0 1) = [ 27 ((sec? 0 _0)+ V(cosec" 0 _ !) = sec € cosec 0 28 (sec? tan? 0) + V(cosec? 0 ~ cot? 0) = 2 cos? 0 29 =1 sin sec? 0 _ 1 sec 0 cosec 0 tan 0 + cot 0 30 tan 0 cot 0 sec 0 + cosec 0 cos 0 sin 0 31 =l V( + tan? 0) V( + cot? 0) Eliminate 0 from the following equations: 32 x = 4 cos 0, y = b sin 0. 33 x = 4 cot 0,y = b cosec 0. 34 x=a tan 0,y =b cos 0. 35 x = ] sin 0, y = [ + cos 0, 36 x =a sec 0,y = b +c cos 0. 37 x=a cosec 0, y = b sec 0. 38 X=l+ tan 0,y = cos 0. 39 x=sin 0 + cos 0,y = sin 0 40 cos X = sCc 0 + tan 0, y = sCc 0 ~ tan 0.

12) There were about 557 principals in attendance. 13) Option A: The college will have about 480 students who prefer cookies. 14) Option C: bar graph, title favorite sport and the x axis labeled sport and the y axis.

The relationship between two quantities that are measured on the same scale and in the same units is known as a percentage. The relative size or magnitude of one thing in comparison to another is represented by proportions, which are frequently stated as ratios, fractions, or percentages. For instance, if there are 8 girls and 12 boys in a class, the girl-to-boy ratio is 8:12, or 2:3, which indicates that there are 3 guys for every 2 girls. Statistics, geometry, and physics are just a few of the mathematical and scientific disciplines that use proportions. Proportions are frequently used in statistics to describe the distribution of a population or a sample.

12) For the number of principals in conference we set up a proportionality with total people as follows:

We know that, 52 principals out of 70 people thus:

52/70 = x/750

Now,

x = (52/70) * 750

x = 557.14

Hence, we can predict that there were about 557 principals in attendance at the conference.

13) Given that, out of 225 students 27 prefer cookies thus,

27/225 = 0.12

That is 12% students like cookies.

Now, for the total number of students we have:

0.12 * 4000 = 480

Thus, option A: The college will have about 480 students who prefer cookies.

14) For the collected data the best representation is option C: bar graph with the title favorite sport and the x axis labeled sport and the y axis labeled number of students, with the first bar labeled basketball going to a value of 17, the second bar labeled baseball going to a value of 12, the third bar labeled soccer going to a value of 27, and the fourth bar labeled tennis going to a value of 44.

15) For the given description of the line plot the median is:

For Bus 47 = 16

For Bus 18 = 13

The faster bus is Bus 47, with a median of 16.

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Annette surveyed a total of 480 people.

Let x be the total number of people Annette surveyed. According to the problem, 37.5% of the people surveyed do not want a new mall, and this amounts to 180 people. We can set up the equation:

0.375 * x = 180

Now, let's solve for x:

x = 180 / 0.375

x = 480

If a cylinder has a volume of 2908.33 in³ and a radius of 11.5 in, the height of the cylinder include the following: C. 7 inches.

In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

By substituting the parameters, we have the following:

2908.33 = 3.14 × 11.5² × h

Height, h = 7.00 inches.

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Let's assume that the original price of the camera is x.

Since Annie sold the camera at a loss of 12%, the selling price is 88% of the original price.

So we can set up the equation:

0.88x = $250

To solve for x, we can divide both sides by 0.88:

x = $250 ÷ 0.88

x = $284.09

Therefore, the original price of the camera was $284.09.

For the mean weight five boxes of cookies, the p-value is equals to 0.04189. If p-value is 0.233 > α, the conclusion is that mean weight of cookies is equals to 16.

We have total five boxes of cookies. Let population of weights is normally distributed, 15.91, 14.07 , 14.88, 16.07, 14.79. Use t.test to determine the p value and hypothesis testing for mean weight of product ( cookies). Let null and alternative hypothesis be defined as

[tex]H_0 : \mu = 16[/tex]

[tex]H_a : \mu<16[/tex]

degree of freedom, df = n - 1 = 5

Using the t-test formula, t value= -2.291,

Using t distribution table and the value p-value for t = -2.2907 and 4 degree of freedom is 0.04189. Hence, required value is 0.04189.

b) Now, p-value = 0.233 for mean weight varied from the specified value of 16.

Level of significance, [tex]\alpha[/tex] = 0.05

As we see p-value = 0.233 > 0.05 ( α), so, this is a significant region and null hypothesis can't be rejected. There is no enough evidence to conclude that the mean weight is actually less than 16 ounces.

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Complete question:

Following are the weights of 5 boxes of cookies, each of which is labeled as containing 16 ounces. Assume that the population of weights is normally distributed, 15.91, 14.07 , 14.88, 16.07, 14.79 . A quality control inspector wants to know whether the mean weight is actually less than 16 ounces. Compute the P-value of the test. Write down your P-value. Also, if he obtained a p-value of 0.233. which conclusion is appropriate?

The ratio of the areas of the bases of the two similar prisms is approximately 26 to 1.

What is the scale factor?

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

Let h1 and h2 be the heights of the two similar prisms, and let A1 and A2 be the areas of their respective bases.

Also, let k be the scale factor that relates the smaller prism to the larger prism.

a. We know that the volumes of the two similar prisms are proportional to the cubes of their corresponding side lengths, so we have:

(kh1)³ / (kh2)³ = 135 / 5000

Simplifying this expression, we get:

h1³ / h2³ = 135 / 5000

Taking the cube root of both sides, we get:

[tex]h1 / h2 = (135 / 5000)^{(1/3)}[/tex]

Using a calculator, we find:

h1 / h2 ≈ 0.405

Therefore, the ratio of the heights of the two similar prisms is approximately 0.405 to 1.

b. The ratio of the areas of the bases of two similar prisms is equal to the square of the scale factor. That is:

A1 / A2 = k²

To find the scale factor k, we can use the fact that the volumes of the two prisms are proportional to k³. Therefore:

k³ = 5000 / 135

k ≈ 5.091

Using this value of k, we get:

A1 / A2 = k² = (5.091)² ≈ 26

Therefore, the ratio of the areas of the bases of the two similar prisms is approximately 26 to 1.

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The equation used to determine the number of days, x, required for the substance to decay to 63 grams is: x ≈ 83.60

To determine the number of days, x, required for a radioactive substance to decay to 63 grams, we can use the exponential decay formula. The equation that represents the decay of a radioactive substance over time is:

N(t) = N₀ * (1/2)^(t/h)

Where:

N(t) is the remaining mass of the substance at time t

N₀ is the initial mass of the substance

t is the time elapsed

h is the half-life of the substance

In this case, we have an initial mass of 475 grams, and we want to find the number of days required for the substance to decay to 63 grams. We can set up the equation as follows:

63 = 475 * (1/2)^(x/20)

To solve for x, we can isolate the exponential term on one side of the equation:

(1/2)^(x/20) = 63/475

Next, we can take the logarithm (base 1/2) of both sides to eliminate the exponential term:

log(base 1/2) [(1/2)^(x/20)] = log(base 1/2) (63/475)

By applying the logarithmic property log(base b) (b^x) = x, the equation simplifies to:

x/20 = log(base 1/2) (63/475)

Finally, we can solve for x by multiplying both sides of the equation by 20:

x = 20 * log(base 1/2) (63/475)

Using a calculator to evaluate log(base 1/2) (63/475) ≈ 4.1802, we find:

x ≈ 20 * 4.1802

x ≈ 83.60

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The correct expressions that give the new acreage are A and C.

A. 0.89y represents the remaining 89% of the original acreage after the sale.

C. y-0.11y represents the original acreage minus the 11% that was sold.

It should be noted that to find the new acreage using the first expression (A), we can simply multiply the original acreage by 0.89:

New acreage = 0.89y

To find the new acreage using the second expression (C), we can subtract 0.11 times the original acreage from the original acreage:

New acreage = y - 0.11y

New acreage = 0.89y

Both expressions give the same result, which is the new acreage after the sale.

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A farmer recently sold a large plot of land. The sale decreased his total acreage by 11%. Let y be the original acreage. Find two equivalent expressions that will give the new acreage. Use pencil and paper. Use the expressions to describe two ways to find the new acreage.

Which two expressions will give the new acreage?

A. 0.89y

OB. 0.11y

C. y-0.11y

D. 11y

OE. y-11y

OF. 0.89y-0.11y

Answer:

127. my answer needs to be 20+ character sooooooooooooo

Answer:

To complete the square for the equation -5x² + 10x = -315, we can follow these steps:

Move the constant term (-315) to the right side of the equation:

-5x² + 10x + 315 = 0

Divide both sides of the equation by the coefficient of the x^2 term (-5) to get a coefficient of -1 for the x^2 term:

x² - 2x - 63 = 0

Add the square of half the coefficient of the x term to both sides of the equation:

x² - 2x + 1 - 1 - 63 = 0 + 1

Simplify the left side of the equation by factoring the quadratic trinomial and combining like terms:

(x - 1)² - 64 = 1

Add 64 to both sides of the equation:

(x - 1)² = 65

So the intermediate step in completing the square for -5x² + 10x = -315 is:

(x - 1)² - 64 = 1

This is the result of adding the square of half the coefficient of the x term (-1) to both sides of the equation and simplifying.

An equation in slope-intercept form for the line shown in the given graph is equal to y = -5x + 5.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (5 - 20)/(0 + 3)

Slope (m) = -15/3

Slope (m) = -5.

At data point (0, 5) and a slope of -5, a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = -5(x - 0)

y - 5 = -5x

y = -5x + 5

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The solution to the given problem of the triangle comes out to be AC are 20 units long.

If a polygon contains at least one more segment, it is a hexagon. It is a simple rectangle in shape. Anything identical to a standard triangle form can only be distinguished from it by edges across A and B. Euclidean geometry does not completely describe the cube, despite the fact that its edges are all collinear. A triangle is formed by a circle with five sides and three angles.

Here,

If DE is the midpoint of triangle ABC, DE is parallel to AC and has a length of 50% of AC.

Let's use "x" units to represent the length of AC.

Given the information, DE equals 10 units, or 50 percent of AC's length. So that we may create the equation:

=>  DE = AC/2

When we enter DE's value as 10 units, we get:

=> 10 = x/2

To separate x, we multiply both sides of the equation by 2 and obtain:

=> 2 * 10 = x

=> x = 20

Therefore, AC is 20 units long.

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For every 100 free throws Chris Paul attempted, he made approximately 93 of them.

We have,

Chris Paul's free throw percentage of 93.4 means that he made 93.4% of his free throw attempts during the 2020-21 season.

In other words, for every 100 free throws he attempted, he made approximately 93 of them.

This is a high free throw percentage and indicates that Chris Paul is a skilled free throw shooter.

Thus,

For every 100 free throws Chris Paul attempted, he made approximately 93 of them.

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