If dy/dt=f(t)g(y), the equilibrium solutions can be obtained by finding the solutions to f(t)=0

Correlation is a statistical measure that describes the strength and direction of a relationship between two variables. It indicates how much one variable tends to change in response to changes in the other variable.

The sample correlation coefficient between X and Y can be calculated using the following formula: r =[tex][nΣxy - (Σx)(Σy)] / [√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2)][/tex]

where n is the sample size, Σxy is the sum of the products of the corresponding x and y values, Σx and Σy are the sums of the x and y values, Σx^2 and Σy^2 are the sums of the squared x and y values, respectively.

Using the given information, we can calculate the necessary values as follows:

n = 45

Σx = 153.7

Σy = 231.2

Σxy = 712.5

Σx^2 = 718

Σy^2 = 1775.2

Substituting these values into the formula, we get:

r = [nΣxy - (Σx)(Σy)] / [√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2)]

r = [45(712.5) - (153.7)(231.2)] / [√(45(718) - (153.7)^2) √(45(1775.2) - (231.2)^2)]

r = 0.804

Therefore, the sample correlation coefficient between X and Y is 0.804. This indicates a strong positive linear relationship between the two variables.

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The statement "The use of the Poisson distribution requires a value n which indicates a definite number of independent trials" is false.

The Poisson distribution is a probability distribution that is used to model the occurrence of rare events in a given time or space interval. It does not require a definite number of independent trials, as it is a continuous probability distribution. Instead, it assumes that the events occur randomly and independently over time or space, with a constant mean rate.

The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of occurrence of the event. Therefore, the Poisson distribution does not require a value n to indicate a definite number of independent trials.

Therefore, the statement "The use of the Poisson distribution requires a value n which indicates a definite number of independent trials" is false.

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The total profit from the sale of the first 60 tickets is -58140.

Given that;

A concert promoter sells tickets and has a marginal profit function of,

⇒ P'(x) = 6x - 1149.

Now, For find the total profit from the sale of the first 60 tickets, we need to integrate the marginal profit function from 0 to 60, which is:

P(x) = ∫[0,60] P'(x) dx ]

P(x) = ∫[0,60] (6x - 1149) dx

P(x) = 3x² - 1149x | from 0 to 60

P(x) = (3(60)² - 1149(60)) - (3(0)² - 1149(0))

P(x) = (10800 - 68940) - 0

P(x) = -58140

Therefore, the total profit from the sale of the first 60 tickets is -58140.

However, it's important to note that a negative profit means the concert promoter is operating at a loss, hence they would need to adjust their pricing strategy or cut costs to remain profitable.

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The function f(x) = log₂(2x² + 2) has an absolute maximum value of log₂(2) ≈ 1 and an absolute minimum value of log₂(2) ≈ 1 on the interval [-1,∞).

To find the absolute maximum and minimum values of f(x) = log₂(2x² + 2) on the interval [-1,∞), we can use the following steps:

Take the derivative of f(x) with respect to x:

f'(x) = 4x / (2x² + 2) ln(2)

Find critical points by setting f'(x) equal to zero and solving for x:

f'(x) = 0 => 4x / (2x² + 2) ln(2) = 0 => x = 0

Check the value of f(x) at the critical point and at the endpoints of the interval:

f(-1) = log₂(2) ≈ 1

f(0) = log₂(2) ≈ 1

As x approaches infinity, f(x) approaches infinity.

Determine the absolute maximum and minimum values of f(x):

The absolute maximum value of f(x) on the interval [-1,∞) is log₂(2) ≈ 1, which occurs at x = -1 and x = 0. The absolute minimum value of f(x) on the interval [-1,∞) is also log₂(2) ≈ 1, which occurs at x = -1 and x = 0.

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Answer:

Step-by-step explanation:

29

Answer:

Based on the information provided, it is not clear what "moms" refers to. If "moms" is meant to represent a unit of measurement or a quantity, the context and units need to be specified for a meaningful calculation. Please provide additional information or clarify your question so that I can provide an accurate response.

29?

To make 6/1 and 12/2 equivalent, we can simplify 12/2 to 6/1 by dividing both the numerator and denominator by 2. This results in two equivalent fractions, 6/1 and 6/1 and show them on number line also.

To make 6/1 and 12/2 equivalent, we can use a number line to represent both fractions and then compare them.

First, we can represent 6/1 on a number line by putting a point at 6 on the line, like

Next, we can represent 12/2 on the same number line by putting a point at 12, which is twice the value of 6, like

Now we can see that both points are on the same line, which means that 6/1 and 12/2 are equivalent fractions.

we can simplify both fractions to a common denominator and compare the resulting numerators. In this case, the common denominator is 2, so we can write

6/1 = 12/2 = 12/2

The numerators of both fractions are equal to 12, which means that 6/1 and 12/2 are equivalent fractions.

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The critical value for the given hypothesis test with a significance level of 0.05 and sample size of 19, testing the alternative hypothesis H1: σ > 4.5, cannot be determined without additional information.

To find the critical value, we need to know the distribution of the data and the desired level of significance (also known as the alpha level) for the hypothesis test. In this case, we are given that the significance level, denoted as alpha (α), is 0.05, but we do not have information about the distribution of the data or the desired level of significance.

The critical value is a value from the distribution that is used as a threshold to determine whether to reject or fail to reject the null hypothesis. If the test statistic (calculated from the sample data) is greater than the critical value, we would reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less than or equal to the critical value, we would fail to reject the null hypothesis.

However, without knowing the distribution of the data and the desired level of significance, we cannot determine the critical value for this hypothesis test. Therefore, we cannot provide a specific numerical value for the critical value in this case.

Therefore, the critical value for the given hypothesis test with a significance level of 0.05 and sample size of 19, testing the alternative hypothesis H1: σ > 4.5, cannot be determined without additional information.

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The number of different sequences of crabapple recipients that are possible in a week are 14,641.

In Mrs. Crabapple's British literature class, there are 11 students, and she gives out a crabapple at the beginning of each of the 4 class meetings per week.
To determine the number of different sequences of crabapple recipients, we will calculate the number of possibilities for each class meeting and multiply them together. Since she can pick any of the 11 students for each class, there are:
11 possibilities for the first class,
11 possibilities for the second class,
11 possibilities for the third class, and
11 possibilities for the fourth class.
So, the total number of different sequences of crabapple recipients in a week is:
11 * 11 * 11 * 11 = 11^4 = 14,641 different sequences.

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Answer:

The answer is No

Step-by-step explanation:

Temperature is measured in K(kelvin),°C(Celsius),°F(Fahrenheit)

The numerical value of a + b + c is 10(option a).

To find the prime factorization of 180, we start by dividing it by the smallest prime number, which is 2. We get:

180 ÷ 2 = 90

So 2 is a prime factor of 180. We continue dividing 90 by 2 until we can no longer divide by 2:

90 ÷ 2 = 45

45 ÷ 2 = 22.5 (not a whole number)

So we move on to the next smallest prime number, which is 3. We divide 45 by 3:

45 ÷ 3 = 15

Now we divide 15 by 3:

15 ÷ 3 = 5

Since 5 is a prime number, we can stop dividing. We have found the prime factorization of 180:

180 = 2 × 2 × 3 × 3 × 5

To express this in the form a²b²c, we group the prime factors in pairs of 2s and 3s:

180 = (2²) × (3²) × 5

Now we can see that a = 2, b = 3, and c = 5. We add them up to get:

a + b + c = 2 + 3 + 5 = 10

Hence the answer is A.

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The math operation which Amanda applied in her first step include the following: C: She subtracted 15 from each side of the equation.

In Mathematics and Geometry, the subtraction property of equality states that the two (2) sides of an algebraic expression or equation would still remain equal even when the same number has been subtracted from both sides of an equality.

By applying the subtraction property of equality to Amanda's equation, we have the following:

30 = 15 - 3x

30 - 15 = 15 - 3x - 15 (first step)

15 = -3x

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The process of a baby picking up a crumb or Cheerio involves several different types of assessment data, including visual, perceptual, fine motor, and proprioceptive skills.

Firstly, the baby uses their visual and perceptual skills to locate the crumb or Cheerio. They may scan the surrounding environment or look directly at the object. This can be assessed through observation of the baby's eye movements and head orientation.

Next, the baby uses their fine motor skills to reach for the crumb or Cheerio. They may use their fingers or their whole hand to grasp the object. This can be assessed through observation of the baby's hand movements and coordination.

Finally, the baby uses their proprioceptive skills to adjust their grip and bring the crumb or Cheerio to their mouth. This can be assessed through observation of the baby's mouth movements and coordination.

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A. A post-lunch nap decreases the amount of time it takes males to sprint 20 meters.

To test if there is enough evidence to support the researcher's claim, we can perform a paired t-test. The null hypothesis is that there is no difference in the mean sprint time between without nap and with nap conditions. The alternative hypothesis is that the mean sprint time is shorter with a post-lunch nap.

(a) Calculate the differences between sprint times with and without nap for each male:
Male      Difference
1               0.01
2               0.01
3               0.02
4               0.01
5               0.017
6               0.06
7               0.01
8               0.03
9               0.01
10             0.03

(b) Calculate the mean difference:
mean difference = 0.022
(c) Calculate the standard deviation of the differences:
s = 0.026
(d) Calculate the t-statistic:
t = (mean difference - 0) / (s / sqrt(n)) = (0.022 - 0) / (0.026 / sqrt(10)) = 2.95
(e) Calculate the degrees of freedom:
df = n - 1 = 9
(f) Determine the critical value for a two-tailed test with alpha = 0.10 and df = 9:
t_critical = +/- 1.833
(g) Compare the absolute value of the t-statistic to the critical value:
|t| = 2.95 > 1.833
(h) The t-statistic falls in the rejection region, so we reject the null hypothesis.
(i) There is enough evidence to support the alternative hypothesis that the mean sprint time is shorter with a post-lunch nap.
(j) The p-value for this test is less than 0.10.
(k) We can conclude with 90% confidence that the mean difference in sprint times with and without nap is between 0.005 and 0.039.
(l) We can conclude with 95% confidence that the mean difference in sprint times with and without nap is between -0.002 and 0.046.
(m) We can conclude with 99% confidence that the mean difference in sprint times with and without nap is between -0.008 and 0.052.
(n) The assumptions for the test are that the samples are random and dependent, and the population is normally distributed.
(o) Based on the results of this test, we can support the researcher's claim that a post-lunch nap decreases the amount of time it takes males to sprint 20 meters after a night with only 4 hours of sleep.

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The rate of change of the surface area of the sphere is -48 π square cm/ sec.

We're given that the radius of a sphere is dwindling at a rate of 2 cm/ sec. Let's denote the sphere's radius by r and the rate of change of the compass by dr/ dt. In this case, dr/ dt = -2( negative because the radius is dwindling).

We're asked to find the rate of change, in square cm/ sec, of the face area of the sphere at the moment when the compass is 3 cm. The face area S of a sphere with radius r is given by the formula S = 4π[tex]r^{2}[/tex].

We can use the chain rule of differentiation to find the rate of change of S with respect to time.

dS/ dt = dS/ dr * dr/ dt

We can find dS dr by secerning the formula for S with respect to r

dS/ dr = 8πr

Now we can substitute r = 3 and dr/ dt = -2 into the expression for dS/ dt

dS/ dt = dS/ dr * dr/ dt

dS/ dt = 8πr *(- 2)( substituting r = 3 and dr/ dt = -2)

dS/ dt = 8π( 3)(- 2)

dS/ dt = -48 π

thus, when the compass of the sphere is 3 cm, the rate of change of the face area of the sphere is -48 π square cm/ sec. The negative sign indicates that the face area is decreasing.

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The minimum distance between the curve and the point (3/2, 0) is [tex]\sqrt (5/4) = \sqrt (5/2)[/tex], which occurs at x = 1.

The curve comes to the point (3/2,0), we need to minimize the square of the distance between the point and the curve.

Let (x, y) be a point on the curve [tex]y = x^3 - 3x + 2[/tex]. Then, the square of the distance between (x, y) and (3/2, 0) is:

[tex]d^2 = (x - 3/2)^2 + y^2[/tex]

Substituting [tex]y = x^3 - 3x + 2[/tex], we get:

[tex]d^2 = (x - (3/2))^2 + (x^3 - 3x + 2)^2[/tex]

To minimize[tex]d^2[/tex], we take the derivative of [tex]d^2[/tex] with respect to x and set it equal to 0:

[tex]d^2/dx = 2(x - (3/2)) + 2(x^3 - 3x + 2)(3x^2 - 3) = 0[/tex]

Simplifying and factoring, we get:

[tex]2(x - (3/2)) + 6(x - 1)(x + 1)(x^2 - x - 1) = 0[/tex]

One solution to this equation is x = 1, which is a local minimum.

Since the curve is symmetric about the y-axis, there is another local minimum at x = -1.

We can check that these are the only two local minima by observing that the second derivative of d^2 is positive at these points.

The curve comes closest to the point (3/2, 0) at x = 1 and x = -1. To find the minimum distance, we substitute these values into the equation for [tex]d^2:[/tex]

[tex]d^2(1) = (1/2)^2 + (1)^2 = 5/4[/tex]

[tex]d^2(-1) = (5/2)^2 + (-3)^2 = 49/4[/tex]

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785 in² is the volume of the cylinder shown .

What is a cylinder, simply defined?

A cylinder is a three-dimensional solid in mathematics that holds two parallel bases spaced at a constant distance apart from one another and connected by a curving surface.

                These bases often have a circular form (like a circle), and a line segment connecting the centres of the two bases is known as the axis.

the volume of the cylinder = πr²h

               r = 10

             h = 25 in

           the volume of the cylinder  = 3.14 * 10 * 25

                                                         = 785 in²

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By using linearization, we can approximate the cube root of 7.9 to be approximately 1.9833.

To use linearization to approximate the value of ∛7.9, we need to first find an appropriate point to use as the basis for our linearization. One common method for choosing this point is to select a value that is close to the desired input, and that simplifies the calculations involved.

In this case, we can choose the point x = 8, which is the nearest perfect cube to 7.9. Evaluating the function at this point, we have f(8) = ∛8 = 2.

Next, we need to find the slope of the tangent line to the function at x = 8. This is given by the derivative of the function at that point. Using the power rule for differentiation.

Evaluating this derivative at x = 8.

Thus, the equation of the tangent line to the function f(x) = ∛x at x = 8 is:

y = f(8) + f'(8)(x - 8)

= 2 + (1/12)(x - 8)

We can now use this linear approximation to estimate the value of ∛7.9. To do this, we substitute x = 7.9 into the equation for the tangent line:

y ≈ 2 + (1/12)(7.9 - 8)

= 2 - (1/120)

= 1.9833...

This is a relatively close approximation to the true value of ∛7.9, which is approximately 1.9834.

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Complete Question:

Use the linearization of f(x) = ∛x at an appropriate point to approximate the value of ∛7.9

Answer: y=0.2x-2

Step-by-step explanation: Find the slope. (4,-1) (0,-2) are the two points I picked.

(-2)-(-1)=(-1)

0-4=(-4)

-1/-4=1/4=0.2

The y-intercept is -2.

Therefore, the answer is y=0.2x-2.

The height of the seedling after one week is approximately 11.695 cm, given that it grew approximately 0.045 cm each day and started at a height of 11.38 cm.

To determine the approximate height of the seedling after one week, we need to calculate how much it will grow in one week and add that to its initial height.

Since the seedling grows approximately 0.045 cm each day, it will grow approximately

0.045 cm/day × 7 days/week = 0.315 cm/week

Therefore, after one week, the height of the seedling will be approximately

11.38 cm + 0.315 cm = 11.695 cm

So the approximate height of the seedling after one week is 11.695 cm.

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By minimizing the total energy, the pigeon should fly at the point where the total energy is minimized. This point is 18.75 miles away from point A.

Energy is the ability to do work. It is the capacity to cause change, move objects, and affect the environment. It is a fundamental part of nature and exists in various forms, such as kinetic energy, potential energy, thermal energy, light energy, chemical energy, and electrical energy.

Point S should be located 18.750 miles away from point A. This is the point where the total energy required to reach point A is minimized. The total energy required is given by:

Total energy = (Energy rate over water) • (Distance over water) + (Energy rate over land) • (Distance over land)

Substituting in the given values, we get:

Total energy = (1.25 * 12) + (1 * 18.75)

Total energy = 24 + 18.75

Total energy = 42.75

By minimizing the total energy, the pigeon should fly at the point where the total energy is minimized. This point is 18.75 miles away from point A.

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The rate of change of the area of the square when the side length is 8 inches and the side length is increasing at a rate of 14 inches per minute is 224 square inches per minute.

To find the rate of change of the area of the square, we need to use the formula for the area of a square:

A = s^2

where A is the area of the square and s is the length of the side of the square.

To find the rate of change of the area, we need to take the derivative of this formula with respect to time:

dA/dt = 2s(ds/dt)

where dA/dt is the rate of change of the area, ds/dt is the rate of change of the side length, and s is the side length of the square.

Since the side length is increasing at a rate of 14 inches per minute, we can substitute ds/dt = 14 into the above equation, and we are given that the side length is 8 inches, so we can substitute s = 8.

dA/dt = 2s(ds/dt)

dA/dt = 2(8)(14)

dA/dt = 224

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To find the mean weight of 31 fruits at random, we can use the Central Limit Theorem. According to the theorem, the sample means of large sample size (n>=30) from any population will be normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The mean weight of 31 fruits at random will be normally distributed with a mean of 451 grams and a standard deviation of 9/sqrt(31) grams.

To find the weight that the mean will be greater than 20% of the time, we need to find the z-score corresponding to the 20th percentile of the normal distribution. Using a standard normal distribution table, we find that the z-score is -0.84.

Now we can use the formula z = (x - mu) / (sigma / sqrt(n)) to find the weight (x) that corresponds to the z-score. Plugging in the values, we get -0.84 = (x - 451) / (9 / sqrt(31)). Solving for x, we get x = 448.4 grams. Therefore, the mean weight of 31 fruits at random will be greater than 448.4 grams 20% of the time.

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Answer: -2

Step-by-step explanation:

In this case, x=-4 from f(-4) and x<-1 so use the first equation

f(-4) =  -4 +2 = -2

The least common multiple (LCM) of the expressions is 9000x⁴y⁵

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

450x²y⁵

3000x⁴y

Factor each expression

So, we have

450x²y⁵ = 2 * 3 * 3 * 5 * 5x²y⁵

3000x⁴y = 2 * 2 * 2 * 3 * 5 * 5 * 5x⁴y

Multiply all factors

So, we have

LCM = 2 * 2 * 2 * 3 * 3 * 5 * 5 * 5x⁴y⁵

Evaluate

LCM = 9000x⁴y⁵

Hence, the LCM is 9000x⁴y⁵

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Answer:

It's difficult to answer this question without the specific equations you are working with. However, in general, if you have an equation involving a variable n and you need to find values of n that are less than 1, you can solve the equation for n and then look for solutions that satisfy the condition.

For example, if you have the equation 2n - 3 = 5, you can solve for n by adding 3 to both sides:

2n - 3 + 3 = 5 + 3

2n = 8

n = 4

In this case, n is not less than 1. However, if you had the equation 0.5n + 2 = 3, you would get:

0.5n + 2 - 2 = 3 - 2

0.5n = 1

n = 2

In this case, n is greater than 1. But if you had the equation 0.5n + 2 = 1, you would get:

0.5n + 2 - 2 = 1 - 2

0.5n = -1

n = -2

In this case, n is less than 1, since it is a negative number.

Answer:

2∣∣∣α1−α2(α1−2)(α2−2)∣∣∣<1for0<α1,α2<1

Step-by-step explanation:

The constant 'a' can be found by solving the integral equation: 2∫₀² 6(2x² + x - a)dx = 24. Simplifying the expression and solving for 'a' gives the value of 5/9.

To find the value of the constant 'a', we need to solve the integral equation:

2∫₀² 6(2x² + x - a)dx = 24

First, we'll integrate the function with respect to x:

2[∫(12x² + 6x - 6a)dx] = 24

Now, we'll find the antiderivative:

2[(4x³/3 + 3x²/2 - 6ax) |₀²] = 24

Next, we'll evaluate the antiderivative at the limits of integration:

2[(4(2³)/3 + 3(2²)/2 - 6a(2)) - (0)] = 24

Simplify the expression:

2[(32/3 + 12 - 12a)] = 24

Divide both sides by 2:

(32/3 + 12 - 12a) = 12

Now, we'll solve for 'a':

-12a = 12 - 12 - 32/3

-12a = -20/3

a = (-20/3) / -12

a = 5/9

So, the value of the constant 'a' is 5/9.

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The Prime factors of 36 are (2²) * (3²) or 2 * 2 * 3 * 3. Thus, option B is the correct answer.

Prime numbers are numbers that are not divisible by numbers other than 1 or the number itself. Composite numbers are numbers that have more than 2 factors other than 1 and the number itself.

Factors are numbers that when divided by another number leave no remainder. Prime factors are the prime numbers that when multiplied the product we get equal to the original number.

To calculate the prime factor, we use the division method.

In this method, firstly we divide the number by the smallest prime number it is when divided by is completely divisible. In this case, we divide 36 by 2 and get 18 as the quotient.

Again, divide the quotient of the previous division by the smallest prime number it is divisible. So, 18 is again divided by 2 and we get 9.

Repetition of the previous step takes place until we get 1. And 9 ÷ 3 = 3. Then  3 ÷ 3 = 1. Since we get 1, this is the final answer.

Finally, Prime factorization of 36 = 2 × 2 × 3 × 3 or we can write it as (2²) * (3²)

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Solving the equation we get, x= -2.

What is equation?

In  algebra, the definition of an equation is a mathematical statement which shows that two mathematical expressions are equal. For example, 3x - 7= 14 is an equation, in which 3x - 7 and 14 are two expressions separated by an 'equal( '=')' sign. Solving the equation we will get the value of the unknown x=7.

Given equation is

                  3/2+2x/5=7/10

    Taking the constants to the right hand side of the equation we get,

                    2x/5= 7/10 - 3/2

         The lowest common denominator of 7/10 and 3/2 is 10

Multiplying by 10 to the both sides of equation we get,

                     (2x/5)×10 = (7/10-3/2)×10

                   ⇒ 4x = (7/10)×10 - (3/2)×10

                   ⇒ 4x = 7- 15

                  ⇒ 4x = -8

         Dividing both sides by 4 we get,

                      x = -2

Hence, solving the equation we get, x= -2.

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The maximum value of f(x,y) = x+y³ for x, y > 0 on the unit circle is (√37 + 2)/9.

We need to find the maximum value of the function f(x,y) = x+y³  on the unit circle, which is the set of all (x,y) points with radius 1 centered at the origin.

Since the domain of the function is restricted to x,y>0, we can use Lagrange multipliers to find the maximum value on the unit circle.

First, we set up the system of equations:

∇f = λ∇g

g(x,y) = x² + y² - 1

Where ∇f and ∇g are the gradient vectors of f and g, respectively, and λ is the Lagrange multiplier.

∇f = <1, 3y²>

∇g = <2x, 2y>

Setting ∇f = λ∇g, we get:

1/2x = λ

3y²/2y = λ

Simplifying, we get:

x = 3y²

Plugging this into the equation of the unit circle, we get:

9y⁴ + y² - 1 = 0

Using the quadratic formula, we get:

y² = (-1 ± √(1 + 36))/18

y² = (-1 ± √37)/18

Since y>0, we take the positive root:

y² = (√37 - 1)/18

Plugging this into x = 3y², we get:

x = 3(√37 - 1)/18

Therefore, the maximum value of f(x,y) = x+y³  on the unit circle is:

fmax = x+y³  = 3(√37 - 1)/18 + (√37 - 1)/54

fmax = (√37 + 2)/9

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