Consider a metal plate on [0,1] ×[0,1] with density rho(x,y) = αx
+ βy g/cm2, where α and β are positive constants. Show that the
center of mass must lie on the line x + y = 7/6 .

Answer:

Step-by-step explanation:

to solve this question, we need to use the graph on the left to find the proportions of students who prefer each pasta dish, and then multiply those proportions by 400 to get the estimated number of students in the whole school who prefer each pasta dish. Then we need to plot those numbers on a bar graph with the pasta dishes on the horizontal axis and the number of students on the vertical axis. The graph below shows one possible way to create the bar graph:

We can see that the vertical axis is numbered from 0 to 120 in increments of 20. The bars show the estimated number of students who prefer each pasta dish, based on the sample of 16 students. For example, since 4 out of 16 students prefer spaghetti, we can estimate that 4/16 x 400 = 100 students in the whole school prefer spaghetti. Similarly, since 3 out of 16 students prefer lasagna, we can estimate that 3/16 x 400 = 75 students in the whole school prefer lasagna. We can repeat this process for the other pasta dishes and plot them on the graph.

✧☆*: .｡. Hope this helps, happy learning! (*✧×✧*) .｡.:*☆

To prove that 5^7 + 5^6 is divisible by 6, we can factor out the common term 5^6 and apply the divisibility rule for 6. Here's the solution:

1. Factor out the common term 5^6:
5^7 + 5^6 = 5^6(5 + 1) = 5^6 * 6

2. Apply the divisibility rule for 6:
A number is divisible by 6 if it is divisible by both 2 and 3. Since 6 is already a multiple of 6 (6*1), the expression 5^6 * 6 is divisible by 6.

Thus, we have proved that 5^7 + 5^6 is divisible by 6.

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The probability of getting two numbers whose sum is 12 when rolling a single number cube twice is 1/36.

To find the probability of getting two numbers whose sum is 12 when a single number cube is rolled twice, we'll first need to identify the favorable outcomes.

A number cube has six faces, numbered from 1 to 6. When it is rolled twice, there are 6 x 6 = 36 equally-likely outcomes. To find the probability, we can use the formula:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

The first step is to identify the favorable outcomes, which are the pairs of numbers that add up to 12. Since we are dealing with a number cube that has faces numbered 1 to 6, there is only one possible pair that satisfies this condition: (6,6).

Now that we have determined the favorable outcome, we can find the probability:

Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 1 (for the pair 6,6) / 36

Thus, the probability of getting two numbers whose sum is 12 when rolling a single number cube twice is 1/36.

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The final exponential function using the values of 'a' and 'b':
y = (1/2)(4^x)

To find an exponential function that passes through the points (2,8) and (4,128), follow these steps:

Step 1: Recall the general form of an exponential function: y = ab^x
Here, 'a' and 'b' are constants that need to be determined using the given points.

Step 2: Substitute the first point (2,8) into the equation:
8 = ab^2

Step 3: Substitute the second point (4,128) into the equation:
128 = ab^4

Step 4: Divide the second equation by the first equation to eliminate 'a':
(128 = ab^4) / (8 = ab^2)
16 = b^2

Step 5: Solve for 'b':
b = √16
b = 4

Step 6: Substitute the value of 'b' back into the first equation:
8 = a(4^2)

Step 7: Solve for 'a':
8 = 16a
a = 8/16
a = 1/2

Step 8: Write the final exponential function using the values of 'a' and 'b':
y = (1/2)(4^x)

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To set up an equation to show the total amount, we can use the formula for compound interest:

A = P(1 + r/n)^nt

Where:

A = the total amount
P = the principal (initial investment)
r = the interest rate
n = the number of times the interest is compounded per year
t = the time period (in years)

For the investment at Citibank:

P = $400
r = 3%
n = 1 (compounded annually)
t = 1 (since it is compounded annually)

So, the equation would be:

A1 = $400(1 + 0.03/1)^(1*1)
A1 = $412

For the investment at the second bank:

P = $400
r = 4.2%
n = ∞ (compounded continuously)
t = 1 (since it is for 1 year)

So, the equation would be:

A2 = $400e^(0.042*1)
A2 = $416.99

To find the total amount, we can add the two amounts together:

Total amount = A1 + A2
Total amount = $412 + $416.99
Total amount = $828.99

Therefore, Talissa's total amount after one year with the two investments is $828.99.

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1. The area of square C is 625 ft². 2. The perimeter of square C is 46 ft. 3. The area of square B is 1456 ft². 4. The length of square A is 4√15 ft.

A key idea in geometry that connects the lengths of a right triangle's sides is known as the Pythagorean Theorem. It says that the hypotenuse's square length, which is the side that faces the right angle, is equal to the sum of the squares of the lengths of the other two sides in a right triangle. It has the following mathematical expression:

a² + b² = c²

1. The area of square C can be found using the Pythagorean Theorem:

a² + b² = c²

Using the area the side is calculates as follows:

Side length of A = √225 = 15 ft

Side length of B = √400 = 20 ft

Now,

15² + 20² = c²

225 + 400 = c²

625 = c²

c = √625 = 25 ft

So the area of square C is:

Area of C = 25² = 625 ft²

2. The perimeter of square C can be found by adding up the side lengths of all three squares:

Perimeter of A = 36 ft, so each side length of A is 9 ft.

Perimeter of B = 48 ft, so each side length of B is 12 ft.

Perimeter of C = Side length of A + Side length of B + Side length of C

Perimeter of C = 9 + 12 + c

We found earlier that c = 25 ft, so we can substitute that in:

Perimeter of C = 9 + 12 + 25 = 46 ft

3. The area of square B can be found using the fact that the areas of squares A, B, and C are related by the equation:

Area of A + Area of B = Area of C

We know the area of A and the area of C, so we can solve for the area of B:

Area of A = 15² = 225 ft²

Area of C = 1681 ft²

Area of A + Area of B = Area of C

225 + Area of B = 1681

Area of B = 1456 ft²

4. Perimeter of B = 64 ft

Side length of B = 64 / 4 = 16 ft

Now we can use the Pythagorean Theorem to find the side length of C:

a² + b² = c²

a = 15 ft (side length of A)

b = 16 ft (side length of B)

c = √(a² + b²) = √(15² + 16²) = √481 = 22 ft

Finally, we can use the Pythagorean Theorem again to find the side length of A:

a² + b² = c²

b = 16 ft (side length of B)

c = 22 ft (side length of C)

a = √(c² - b²) = √(22² - 16²) = √240 = 4√15 ft

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1. It's impossible to determine exactly when Lin's battery will die without more information on her phone's battery capacity and usage patterns. However, we can estimate that it will likely die sometime after noon if the rate of battery drain remains constant.

2. Yes, battery life is a function of time. The longer a battery is in use, the more its charge will deplete. However, it is not necessarily a linear function as the rate of battery drain can vary depending on factors such as usage patterns, app activity, and temperature. In this specific case, it appears that the rate of battery drain may be slowing down as the percentage decrease from 90% to noon is less than the decrease from fully charged to 90%.

In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used. A linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input.

Linear functions are related to linear equations.

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Tabitha could pay $4,408.29 less for her SUV if she takes the loan for five years in place of 7.

To calculate the total amount Tabitha might pay for the SUV with a 7-year loan, we are able to use the compound interest formula for calculating the total amount of a loan, that is:

[tex]total amount = principal x (1 + interest charge)^{time}[/tex]

Wherein:

With a 7-year loan, the full amount Tabitha could pay is:

[tex]total amount = $29,000 x (1 + 0.043)^{7} = $37,501.76[/tex]

To calculate the overall amount she could pay with a 5-year loan and a 5% decrease interest rate, we first need to calculate the new interest price. A 5% discount inside the interest price of 4.3% is:

New interest price = 4.3% - 5% = 3.3%

Then we can use the equal formula as before to calculate the total amount with the 5-year mortgage:

[tex]total amount = $29,000 x (1 + 0.033)^{5} = $33,093.47[/tex]

The difference in the overall amount among the two loans is:

$37,501.76 - $33,093.47 = $4,408.29

Therefore, Tabitha could pay $4,408.29 less for her SUV if she takes the loan for five years in place of 7.

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The gross income for selling 348 bushels of apples at a price of $16.50 per bushel is $5,742.

The gross income is the total revenue earned from the sales of a product or service, before any expenses or deductions are taken out. To calculate gross income, we multiply the quantity of goods sold by the price per unit.

In this case, we are given that 348 bushels of apples were sold at a price of $16.50 per bushel. Multiplying these two values together gives us the total revenue earned from the sale of these apples, which is the gross income.

To calculate the gross income, we can use the formula:

Gross income = Quantity sold x Price per unit

Plugging in the given values, we get:

Gross income = 348 x $16.50

Simplifying the calculation, we get:

Gross income = $5,742

Therefore, the gross income for selling 348 bushels of apples at a price of $16.50 per bushel is $5,742.

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In a binomial experiment with n = 10 and p = 0.40 there is a 57.0% chance of getting 5 or more successes in the 10 trials.

A binomial experiment is a statistical experiment that consists of a fixed number of independent trials, where each trial can have only two outcomes, typically called "success" or "failure." The probability of success for each trial is denoted by p, and the number of trials is denoted by n.

In this case, we are given n = 10 and p = 0.40. This means that we are conducting an experiment with 10 independent trials, where the probability of success for each trial is 0.40.

Using this information, we can answer questions about the probability of various outcomes. For example, we can calculate the probability of getting exactly 5 successes in the 10 trials:

P(X = 5) = (10 choose 5) * 0.40^5 * (1 - 0.40)⁽¹⁰⁻⁵⁾

P(X = 5) = 0.246

This means that there is a 24.6% chance of getting exactly 5 successes in the 10 trials.

We can also calculate the probability of getting 5 or more successes:

P(X >= 5) = P(X = 5) + P(X = 6) + ... + P(X = 10)

P(X >= 5) = 0.246 + 0.204 + 0.088 + 0.026 + 0.005 + 0.001

P(X >= 5) = 0.570

This means that there is a 57.0% chance of getting 5 or more successes in the 10 trials.

Overall, the binomial distribution is a useful tool for modeling situations where there are a fixed number of trials with a binary outcome, and the probability of success is known for each trial.

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The area of triangle HGF is 9.8 cm²

The Sides of the given triangle are 6.5 cm, 3.6 cm and 5.6 cm

We know that the area of the triangle is the half of product of base and height of the triangle.

The base of the triangle = 6.5 cm

Height of the triangle = 3.0 cm

We know that the triangle is

Area = 1/2 × base × height

= 1/2 × HF × GE

= 1/2 × 6.5 × 3.0

= 1/2 × 19.5

= 9.75

Rounding to the nearest tenth

= 9.8 cm²

Hence, the area of triangle HGF is 9.8 cm²

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Given question is incomplete, the complete question is below

What is the area of triangle HGF? Round your answer to the nearest tenth of a square centimeter. Recall that you need to round up if the value of the hundredth is 5 or greater.

To make a number line for the values of x that are less than -1 and greater than 1, we can start by drawing a horizontal line and marking a point at 0. Then, we can label the left side of the line with negative numbers and the right side with positive numbers.

Next, we need to mark all the points that represent the values of x that satisfy the condition X < -1 and x > 1. This means we are looking for all the numbers that are less than -1 and greater than 1 at the same time. However, there are no numbers that satisfy this condition since a number cannot be both less than -1 and greater than 1 simultaneously.

Therefore, there are no points to mark on the number line for this condition.

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Based on the given options, the digital text feature that plays music or speaking on its own is "sound." The correct option is sound.

Digital texts can incorporate various features to engage readers and enhance their experience. These features include:

1. Animation: Visual effects and motion graphics that add dynamic elements to a digital text. Animation can be used to illustrate complex concepts, tell stories, or create visual appeal.

2. Interactive elements: Components of digital texts that allow users to interact with the content, such as hyperlinks, buttons, quizzes, or navigation tools. Interactive elements can make digital texts more engaging and help users find information more efficiently.

3. Sound: Audio elements that play music, speaking, or other sounds. Sound can be used to provide additional information, create atmosphere, or guide users through a digital text.

4. Text: The written content of a digital text. Text can be formatted in various ways, such as using different fonts, colors, or layouts, to make digital texts more accessible and visually appealing.

In this case, the feature that plays music or speaking on its own is "sound." Sound elements in digital texts can enhance the user experience by providing audio feedback, creating a more immersive environment, or adding an additional layer of information.

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The area of the King-sized mattress is 4,800 square inches and the area of the Queen-sized mattress is 3,200 square inches.

To find the dimensions of the Queen-sized mattress, let x be the width. Then, the length of the mattress is x + 16. The area of the Queen-sized mattress is x(x + 16) = x² + 16x.

Since the King-sized mattress has the same length as the Queen-sized mattress, its length is x + 16. Also, its width is 20 inches wider than the Queen-sized mattress, so its width is x + 16 + 20 = x + 36. Therefore, the area of the King-sized mattress is (x + 16)(x + 36) = x² + 52x + 576.

The area of the King-sized mattress is 1,600 square inches larger than the area of the Queen-sized mattress. So, we can set up the following equation:

x² + 52x + 576 - (x² + 16x) = 1600

Simplifying this equation, we get:

36x + 576 = 1600

36x = 1024

x = 28

Therefore, the width of the Queen-sized mattress is 28 inches and the length is 44 inches. So, its area is 3,200 square inches.

The width of the King-sized mattress is 28 + 36 = 64 inches and the length is 44 inches. So, its area is 4,800 square inches.

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The diameter of the sphere is approximately 20 meters to the nearest tenth of a meter.

The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere. Since we are given the volume of the sphere, we can solve for the radius as follows:

V = (4/3)π[tex]r^{3}[/tex]

[tex]r^{3}[/tex] = (3V) / (4π)

r = [tex](3V/4\pi )^{1/3}[/tex]

Substituting the given value of the volume, we get:

r = [tex](3*7483/4\pi )^{1/3}[/tex] ≈ 10.0

Therefore, the radius of the sphere is approximately 10 meters. The diameter of the sphere is twice the radius, so the diameter is approximately:

2 x 10 ≈ 20 meters

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The part of the surface area equation are

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

SA = B + 1/2Pl

In the equation, we have

Part 1 = B

Part 2 = P

Part 3 = l

When the above parts are labelled, we have

Part 1 = B = Base area

Part 2 = P = Perimeter of base

Part 3 = l = Height of pyramid

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To find the length of the parametric curve, we can use the formula:

L = ∫a^b √(dx/dt)² + (dy/dt)² dt

Substituting the given values, we get:

L = ∫0^1 √(6)² + (43/2, 03752)² dt

Simplifying:

L = ∫0^1 √(36 + (43/2, 03752)²) dt

Therefore, the length of the parametric curve is:

L = √(36 + (43/2, 03752)²)

In mathematics, substitution refers to the process of replacing a variable or expression in an equation or formula with another variable or expression that has the same value.

For example, if we have an equation: 2x + 3y = 7, and we want to substitute x with the value 4, we replace x with 4 to get:

2(4) + 3y = 7

8 + 3y = 7

We can then solve for y to find its value.

Substitution is a commonly used technique in algebra and calculus to simplify expressions, solve equations and evaluate functions.

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Step-by-step explanation:

To solve for inverse, utilize the following steps.

Step 1: let f(x)=y so we get

[tex]y = \sqrt{x - 6} + 5[/tex]

Step 2: Swap y and x

[tex]x = \sqrt{y - 6} + 5[/tex]

Solve for y.

[tex]x - 5 = \sqrt{y - 6} [/tex]

[tex](x - 5) { }^{2} + 6 = y[/tex]

Step 4: Let y =f^-1(x)

[tex](x - 5) {}^{2} + 6 = f {}^{ - 1} (x)[/tex]

Answer: [tex]f^{-1}(x) =[/tex] x²-10x+19

Step-by-step explanation:

Let's replace f(x) for y for now.

[tex]y=\sqrt{x-6}+5[/tex]

To find inverse.  make your y into x, and your x into y

[tex]x=\sqrt{y-6}+5[/tex]        >Now you solve for y.  subtract 5 from both sides

[tex]x-5=\sqrt{y-6}[/tex]        >Square both sides to get rid of root

[tex](x-5)^{2} =(\sqrt{y-6})^{2}[/tex]     >drop root and square (x-5)

(x-5)(x-5) = y-6              >FOIL

x²-5x-5x+25 = y-6        > combine like terms

x²-10x+25 = y-6            >add 6 to both sides

x²-10x+19=y            > this is your inverse now put the y into inverse form

[tex]f^{-1}(x) =[/tex] x²-10x+19

[tex]\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=11\\ \theta =60 \end{cases}\implies s=\cfrac{(60)\pi (11)}{180}\implies s=\cfrac{11\pi }{3}\implies s\approx 12~mm[/tex]

Answer: 12mm

Step-by-step explanation:

Basically, you will find the circumference of the entire circle and then using that find the length of the arc.

So the circumference of the circle is its radius (11) times pi multiplied by 2.

2(11 x 3.14) = 69.08

Now a circle is always 360 degrees and the angle of the sector is 60 degrees.

So we have our circumference and we only need that small portion, so you take and make it a fraction and multiply by the circumference to find the length of that small portion:

60/360 x 69.08 = 11.51

Rounded = 12

189*365=68985

68985/15= $4599

Step-by-step explanation:

or $5000 if you round up

Alex will have to put money in his savings account for 15 more weeks at a rate of $8 per week.

Alex purchased a cell phone for $200 using money from his savings account. He plans to replace the money in his savings account at the rate of $8 per week. He has already replaced $80, which means he needs to replace $120 more to have the entire amount. Since he is replacing the money at a rate of $8 per week, he will need 15 more weeks to replace the remaining amount. Therefore, he will have to put money in his savings account for 15 more weeks to replace the entire $200 he spent on the cell phone. By doing so, he will be able to maintain his savings and avoid any financial loss.

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Answer:

Supplementary angles are angles that have a sum of 180°. When two angles are supplementary, we say that one angle is the supplement of the other.

Complementary angles are angles that add up to 90°. When two angles are complementary, we can say that one angle is the complement of the other

Step-by-step explanation:

The correct response is 12, as finding the predicted smoking and asbestos exposure group requires finding x, and in that case, the ratios in the rows and columns are equal.

4/2 = x/6 Or 6/2= x/4

This suggests that x = 12.

Smoking refers to the act of inhaling and exhaling the smoke produced by burning tobacco or other substances such as marijuana or hookah. It is a highly addictive habit that poses significant health risks to both smokers and those exposed to secondhand smoke. Smoking can lead to a variety of illnesses and diseases, including lung cancer, heart disease, stroke, chronic obstructive pulmonary disease (COPD), and various other cancers. It is estimated that smoking is responsible for nearly 8 million deaths globally each year.

The chemicals in tobacco smoke can also harm the environment, contributing to air pollution and litter. Despite the well-known risks associated with smoking, many people continue to smoke due to addiction, peer pressure, or a lack of understanding about the long-term consequences.

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Answer:

To find the weighted mean, we need to multiply each delivery value by its corresponding frequency, add the products, and divide by the total frequency.

(3 x 7) + (6 x 6) + (9 x 1) + (12 x 0) = 21 + 36 + 9 + 0 = 66

Total frequency = 7 + 6 + 1 + 0 = 14

Weighted mean = 66 / 14 = 4.7 (rounded to the nearest tenth)

Therefore, the weighted mean is 4.7

La altura del mástil es de aproximadamente 20.87 metros.

Para resolver el problema, se puede utilizar la ley de cosenos para encontrar la longitud del mástil:

c² = a² + b² - 2ab cos C

Donde:

c es la longitud del mástil (lo que se busca)

a es la longitud de la parte delantera del barco (25m)

b es la longitud de la parte trasera del barco (también 25m)

C es el ángulo entre a y b, que se puede calcular utilizando la tangente: tan C = 1.5 (pues tan 60° = √3, y tan 45° = 1)

Resolviendo para c:

c² = 25² + 25² - 2(25)(25)cos(arctan 1.5)

c = √(25² + 25² - 2(25)(25)(1/√(1+(1.5)²)))

c ≈ 31.08 m

Por lo tanto, la altura del mástil es de aproximadamente 31.08 metros.

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Kate's mistake was in Step 2: "Divide the volume of Wesley's tank by 2." Since she wants a tank with half the volume of Wesley's tank, she should have divided the volume of Wesley's tank by 2, not the radius.

The correct approach would be to use the formula for the volume of a sphere:

V = (4/3)πr^3

Since Wesley's tank has a diameter of 30 cm, its radius is 15 cm. Substituting this value into the formula gives:

V = (4/3)π(15)^3

V ≈ 14,137 cm^3

To find the radius of the smaller tank with half the volume, Kate should have solved for the radius using the formula for the volume of a sphere:

(4/3)πr^3 = (1/2)(4/3)π(15)^3

Simplifying this equation gives:

r^3 = (1/2)(15)^3

r ≈ 10.8 cm

Therefore, the approximate diameter of the smaller tank should be 21.6 cm (twice the radius), not 10.8 cm.

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The value of c that makes the two expressions equivalent is -20.74.

Let's start by simplifying the expression 6.1(2x - 3.4). To do this, we use the distributive property of multiplication, which states that a(b + c) = ab + ac. Applying this to our expression, we get:

6.1(2x - 3.4) = 6.1(2x) - 6.1(3.4) = 12.2x - 20.74

Now we can rewrite the equation we want to solve as:

12.2x + c = 12.2x - 20.74

To solve for c, we need to isolate it on one side of the equation. We can do this by subtracting 12.2x from both sides:

c = -20.74

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For the function f(x) = x^2e^7x the critical numbers are 0 and -2/7 and using the First Derivative Test local minima at x = 0 and -2/7.

The function f(x) = x²e⁷ˣ

Differentiate the function f(x) with respect to x

f'(x) = 2xe⁷ˣ + 7x²e⁷ˣ

f'(x) = xe⁷ˣ(2 + 7x)

Now put f'(x) = 0 for critical points.

xe⁷ˣ(2 + 7x) = 0

x = 0                 2 + 7x = 0              e⁷ˣ ≠ 0

x = 0                 x = -2/7                e⁷ˣ ≠ 0

Hence, 0 and -2/7 are the critical points.

f'(x) = 2xe⁷ˣ + 14x²e⁷ˣ

Now f''(x) = 2e⁷ˣ + 14xe⁷ˣ + 28xe⁷ˣ + 98x²e⁷ˣ

f''(x) = 2e⁷ˣ + 42xe⁷ˣ + 98x²e⁷ˣ

f''(x) = 2e⁷ˣ(1 + 21x + 49x²)

Now f''(x) at x = 0

f''(0) = 2e⁷⁽⁰⁾(1 + 21(0) + 49(0)²)

f''(0) = 2(1 + 0 + 0)

f''(0) = 2 > 0

So f(x) has minima at x = 0

Local minima at x = 0 and y = 0.

Now f''(x) at x = -2/7

f''(0) = 2e⁻²(1 + 21(-2/7) + 49(-2/7)²)

f''(0) = 2e⁻²(1 - 6 + 4)

f''(0) = -2e⁻² < 0

f(x) has local minima at x = -2/7

Now y at x = -2/7 is

y = f(-2/7)

y = (-2/7)² e⁻²

y = 4/49e²

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The equation y = (4/5)x - 3 is in slope-intercept form, y = mx + b, where m is the slope of the line. Therefore, the slope of the line represented by the equation is:

m = 4/5

So the answer is C) 4/5.

Answer: d

Step-by-step explanation:

Answer:

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