Use the properties of logarithms to simplify as much as possible. 3) In(4x^5) – In (x^3)- In 4 4) The price of beef has inflated by 2%. If the price of beef inflates 2% compounded biannually, how lung will it take for the price of beef to triple?

The probability of picking a 20 dollar bill is 1/7 or 14.3%

The probability of a thing is the likelihood or number of chances that such a thing will occur. For the scenario given,

There are a total of 7 bills in your pocket

1, 1, 1, 1,

5, 5,

20.

To find the probability of picking out the twenty dollar bill, divide the number of twenty dollar bills by the the total of the number of bills you are with.

Probability = 1/ 7 which can be converted to % = 14.3%.

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The probability that four boys are given the first four seats on the first teacup is approximately equal to 0.004.

To determine the probability that four boys are given the first four seats on the first teacup.

The total number of ways to distribute 12 tickets among 12 seats is 12! (12 factorial), which is equal to 479,001,600.

We need to find the number of ways that four boys can be selected from the seven boys, multiplied by the number of ways that eight people (including the remaining three boys and five girls) can be selected from the ten remaining people,

multiplied by the number of ways that the selected people can be arranged on the teacup ride.

The number of ways to select four boys from seven boys is 7C4, which is equal to 35. The number of ways to select eight people from the remaining ten people is 10C8, which is equal to 45.

Finally, the number of ways to arrange the selected twelve people on the teacup ride is 4!, which is equal to 24.

Therefore, the probability that four boys are given the first four seats on the first teacup is (35 x 45 x 24) / 12!, which is approximately equal to 0.004.

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The component of b onto a is (-1/3, 2/3, -1/3).

To find the component of b onto a, we first need to find the projection of b onto a. The projection of b onto a is given by the formula:

proj_a(b) = (b dot a / ||a||^2) * a

where dot represents the dot product and ||a|| represents the magnitude of vector a.

We can calculate the dot product of a and b as follows:

a dot b = (-2*1) + (4*0) + (2*3) = 4

We can calculate the magnitude of a as follows:

||a|| = sqrt((-2)^2 + 4^2 + 2^2) = sqrt(24) = 2sqrt(6)

Now we can plug these values into the formula for the projection of b onto a:

proj_a(b) = (b dot a / ||a||^2) * a
proj_a(b) = (4 / (2sqrt(6))^2) * (-2, 4, 2)
proj_a(b) = (4 / 24) * (-2, 4, 2)
proj_a(b) = (-1/3, 2/3, -1/3)

Finally, the component of b onto a is simply the projection of b onto a:
comp_a(b) = (-1/3, 2/3, -1/3)
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Answer:

C

Step-by-step explanation:

All the other ones aren't increasing with the same proportion

Only C is increasing by same number each time (which is 26)

The expression that is equivalent to the one Regina wrote is y + 27/4

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

y + 9 x 3/4

This means that

Expression = y + 9 x 3/4

Expanding the above expression, we have

Expanded expression = y + 27/4

Using the above as a guide, we have the following:

The expression that is equivalent to the one Regina wrote is y + 27/4

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The solution set for the equation 1/2 = 1/3 - |(x-3/6)+x| is {1/12, -1/12}.

To solve the equation 1/2 = 1/3 - |(x-3/6)+x|, we need to isolate the absolute value expression and solve for x in two cases.

Case 1: (x-3/6)+x is nonnegative, which means the absolute value can be removed.

1/2 = 1/3 - (x-3/6)-x

1/2 = 1/3 - 2x + 1/2

2x = 1/6

x = 1/12

Case 2: (x-3/6)+x is negative, which means the absolute value must be flipped.

1/2 = 1/3 + (x-3/6)+x

1/2 = 1/3 + 2x - 1/2

2x = -1/6

x = -1/12

Therefore, the solution set is {1/12, -1/12}.

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The percent of the phosphorus-32 decays each day is 0.56%.

The formula for the amount of radioactive isotope remaining after t days is given as:

y = a(0.5)^(t/14)

To find the percent of phosphorus-32 that decays each day, we need to find the fraction of the initial amount that decays each day. This can be found by subtracting the amount remaining after one day from the initial amount, and then dividing by the initial amount:

fraction decayed in one day = (a - a(0.5)^(1/14)) / a

Simplifying this expression gives:

fraction decayed in one day = 1 - (0.5)^(1/14)

To find the percent decayed in one day, we multiply by 100:

percent decayed in one day = 100(1 - (0.5)^(1/14))

Using a calculator, we get:

percent decayed in one day ≈ 0.56%

Therefore, the percent of phosphorus-32 that decays each day is approximately 0.56%.

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According to the information the length of the ladder to reach the hoop will be approximately 10.77 feet.

How to calculate the length of the ladder?

 Analyzing the problem, we can see that the ladder, the height and the distance from the base of the basket will form a right triangle. We can then use the Pythagorean theorem to calculate the length of the ladder, which will be the hypotenuse of the triangle. The formula used will be:

Substituting the information in the formula we have:

So let's use the square root of 116 to find how long the ladder must be to reach the hoop, which in this case will be:

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The answer is (d) –0.9568, –0.9659.

To find the first five terms of the trigonometric series for cos(4π/7), we can use the formula:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Substituting x = 4π/7, we get:

cos(4π/7) = 1 - (4π/7)²/2! + (4π/7)⁴/4! - (4π/7)⁶/6! + (4π/7)⁸/8!

Using a calculator to evaluate each term and rounding to four decimal places, we get:

cos(4π/7) ≈ -0.9568

Comparing this approximation to the actual value of cos(4π/7), which is approximately -0.9659, we see that the approximation is fairly close but not exact. So, the answer is (d) –0.9568, –0.9659.

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The probability that a student gets a lunch with chips and apple juice is 1/12, and the probability that a student gets a lunch without chips is 1/2.

There are 3 choices of sandwiches, 2 choices of sides, and 2 choices of drinks, so there are a total of 3x2x2 = 12 possible lunch combinations. To find the probability that a student gets a lunch that includes chips and apple juice, we need to count the number of lunch combinations that include chips and apple juice, and then divide by the total number of possible lunch combinations.

Number of lunch combinations that include chips and apple juice = 1 (chips and apple juice is only one combination)

Total number of possible lunch combinations = 12

Probability of getting a lunch that includes chips and apple juice = 1/12

To find the probability that a student gets a lunch that does not include chips, we need to count the number of lunch combinations that do not include chips, and then divide by the total number of possible lunch combinations,

Number of lunch combinations that do not include chips = 6 (3 choices of sandwiches x 2 choices of drinks) Total number of possible lunch combinations = 12, probability of getting a lunch that does not include chips = 6/12 = 1/2

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To find the first term and common ratio of a geometric sequence, we can use the formula for the nth term:

a_n = a_1 * r^(n-1)

We are given the eighth and eleventh terms, so we can set up two equations:

a_8 = a_1 * r^(8-1) = 81920

a_11 = a_1 * r^(11-1) = 5242880

After dividing the second with by the first equation, we get:

(a_1 * r^(11-1)) / (a_1 * r^(8-1)) = 5242880 / 81920

Simplifying, we get:

r³ = 64

Doing the root of cube both sides, we get:

r = 4

Substituting this into the first equation, we get:

a_1 * 4^(8-1) = 81920

a_1 * 4^7 = 81920

a_1 = 5

Therefore, the first term is 5 and the common ratio is 4.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r). The formula for the nth term of a geometric sequence is:

an = a * r^(n-1)

Given that the 8th term (a8) is 81,920 and the 11th term (a11) is 5,242,880, we can set up the following equations:

81920 = a * r^(8-1) => 81920 = a * r⁷ (1)

5242880 = a * r^(11-1) => 5242880 = a * r¹⁰ (2)

Now, we need to find the values of a (the first term) and r (the common ratio). Divide equation (2) by equation (1):

(5242880 / 81920) = (a * r¹⁰) / (a * r⁷)

64 = r^3

Now, we can find the common ratio r:

r = 4 (since 4³ = 64)

Next, substitute r back into equation (1) to find the first term a:

81920 = a * 4⁷

a = 81920 / 16384

a = 5

So, the first term (a) is 5, and the common ratio (r) is 4.

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By using  sum , difference , product , quotient and one that involves at least two operations that have the value of -3/4.

Now, We have to find the five expressions :

Expression for sum is -

1/4 + (-1) = -3/4

Expression for difference is -

(1/4 - 1) = -3/4

Expression for product is -

(-3/2)(1/2) = -3/4

Expression for quotient is -

(1 - 4)/4 = -3/4

Expression that involves at least two operations is -

-(1/4 + 2/4) = -3/4

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a) The profit at a sales level of 14 trees is $180.40.

b) The average change in profit sales is $13.96 per tree [(ΔP/Δx) = 97.75/7].

c) The instantaneous rate of change of profit at a sales level of 14 trees is $19.72 per tree.

d) The equation of the tangent line at (5, f(5)) is y = 6x - 25.

a. To find the profit at a sales level of 14 trees, we need to evaluate the profit function at x = 14:

P(x) = 20x - 0.01x^2 - 100

P(14) = 20(14) - 0.01(14)^2 - 100 = $180.40

Therefore, the profit at a sales level of 14 trees is $180.40.

b. To find the average change in profit sales from 12 to 19 trees, we need to calculate the average rate of change of the profit function over this interval:

Δx = 19 - 12 = 7

ΔP = P(19) - P(12) = (2019 - 0.0119^2 - 100) - (2012 - 0.0112^2 - 100) = $97.75

Therefore, the average change in profit sales is $13.96 per tree [(ΔP/Δx) = 97.75/7].

c. To find the instantaneous rate of change of profit at a sales level of 14 trees, we need to find the derivative of the profit function at x = 14:

P(x) = 20x - 0.01x^2 - 100

P'(x) = 20 - 0.02x

P'(14) = 20 - 0.02(14) = $19.72

Therefore, the instantaneous rate of change of profit at a sales level of 14 trees is $19.72 per tree.

d. To find the equation of the tangent line at (5, f(5)), we need to find the slope of the tangent line and its y-intercept:

f(x) = x^2 - 4x

f'(x) = 2x - 4

f'(5) = 2(5) - 4 = 6

The slope of the tangent line at (5, f(5)) is 6.

To find the y-intercept of the tangent line, we can use the point-slope form of a line:

y - f(5) = m(x - 5)

y - (5^2 - 4*5) = 6(x - 5)

y - 5 = 6x - 30

y = 6x - 25

Therefore, the equation of the tangent line at (5, f(5)) is y = 6x - 25.

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To determine this family's annual medical aid tax credit, we need to consider their medical aid expenses for the year. Medical aid expenses are expenses related to medical services that are not covered by the government or medical insurance.

This family can claim a tax credit of up to R310 per month for the main member and the first dependent, and R209 per month for each additional dependent. This tax credit is only applicable to registered medical schemes and is deducted from the tax payable.

In addition to medical aid expenses, this family will also need to consider the 15% VAT payable on all goods and services, except for sanitary pads, fresh produce, and a few other staple food items. This means that if this family spends R10,000 on goods and services, they will need to pay an additional R1,500 in VAT.

However, the good news is that they won't have to pay VAT on their fresh produce and staple food items. This will help to reduce their overall expenditure on food, which is an essential expense for every family.

In conclusion, while this family will need to pay VAT on most goods and services, they can claim a tax credit for their medical aid expenses. Additionally, they won't have to pay VAT on fresh produce and staple food items, which will help to reduce their overall food expenditure.

By carefully managing their expenses and taking advantage of tax credits and exemptions, this family can ensure that they are able to provide for their essential needs while also managing their financial obligations.

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Answer:

x = 4

Step-by-step explanation:

10 - 4x + 6 - 2x = -2x

10 - 6x + 6 = -2x

16 - 6x = -2x

16 - 4x = 0

-4x = -16

x = 4

Answer:

x = 4

Step-by-step explanation:

Add like terms

-6x + 16 = -2x

Bring like terms to the opposite side

16 = 4x

Divide both sides by 4

x = 4

The given system of equations has one solution.

To determine the number of solutions of the functions. The given system over the interval (-3, 1], we need to find the intersection points of the two functions, f(x) and g(x), within that interval.

First, let's analyze each function separately:

The natural logarithm function ln(x) is only defined for positive                  values of x. In this case, we have ln(x + 3). To find the intersection points              with the interval (-3, 1], we need to ensure that x + 3 is positive.

For x in the interval (-3, 1], we have:

-3 < x ≤ 1

Adding 3 to both sides of the inequality:

0 < x + 3 ≤ 4

Therefore, the function f(x) = ln(x + 3) is defined over the interval (0, 4].

    2. Function g(x) = 2 * [tex]6^x[/tex]:

The exponential function [tex]6^x[/tex] is always positive for any real value of x. Multiplying it by 2 won't change the fact that the function remains positive. Hence, g(x) is positive for all real values of x.

Now, let's determine the intersection points of f(x) and g(x) within the interval (-3, 1].

Since g(x) is always positive and f(x) is defined over (0, 4], the intersection points occur where f(x) = g(x) > 0.

To solve this equation, we can rewrite it as ln(x + 3) - 2 * [tex]6^x[/tex] = 0.

Finding the exact solutions to this equation is not straightforward and may require numerical methods or graphing. However, it's clear that there is at least one solution within the interval (0, 4].

In conclusion, the given system has at least one solution over the interval (-3, 1].

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The volume of a trapezoidal prism is 2362.5.

The total surface area​ is 852.

We have,

The volume of a trapezoidal prism.

V = ((a + b) / 2) × h × l

where:

a and b are the lengths of the two parallel sides (the bases) of the trapezoid

h is the height of the trapezoid (the perpendicular distance between the two bases)

l is the length of the prism (the distance between the two trapezoidal faces)

Now,

a = 9

b = 12

l = 15

Height h can be calculated using the Pythagorean theorem.

15² = (12 - 9)² + h²

h² = 225 + 9

h² = 234

h = √234

h = 15

Now,

The volume of a trapezoidal prism.

V = ((a + b) / 2) × h × l
V = ((9 + 12) / 2) x 15 x 15

V = 2362.5

And,

The surface area (A) of a trapezoidal prism can be calculated using the formula:

A = ph + 2B

where p is the perimeter of the trapezoidal base, h is the height of the prism, and B is the area of one of the bases.

So,

p = 12 + 8 + 12 + 8 = 44

h = 15

B = 12 x 8 = 96

Now,

Total surface area​.
= 44 x 15 + 2 x 96

= 660 + 192

= 852

Thus,

The volume of a trapezoidal prism is 2362.5.

The total surface area​ is 852.

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The equation that represents the Ferrells' savings after x months is y = 150x

In this equation, x represents the number of months that the Ferrells have been saving, and y represents the amount of money they have saved after x months.

The coefficient 150 represents the amount of money the Ferrells save each month, and it is multiplied by the number of months x to get the total savings y. For example, after 5 months of saving, the Ferrells would have saved:

y = 150(5) = $750

This equation can be used to calculate their savings at any point in time, as long as they continue to save $150 each month.

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176, 264. your welcome!

The slope of the line represent Parking cost $30 per day. The correct answer is C.

The slope of a line represents the rate of change between two variables. In this case, the line represents the relationship between Hector's parking fees and the number of days he parked his SUV.

The unit of the slope of the ine represents

Cost/ number of days = 30 $/day

The fact that the slope is negative (-$30) means that for each additional day Hector parked his SUV, his parking fees decreased by $30. This indicates that the parking fee is a function of the number of days parked, and it costs $30 per day. The correct option is C.

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--The given question is incomplete, the complete question is given

" Hector parked his SUV in long-term parking while he traveled to China. What does the slope of the line represent?

A Hector's parking fees decreased by $30 each week.

B Parking cost a flat fee of $30.

C Parking cost $30 per day.

D Hector got a $30 discount to park his SUV. ​ "--

If the parallelogram has an area of 25.2 cm² and the height is 4 cm, the length of the base is 6.3 cm.

To start, we know that the area of a parallelogram is given by the formula:

A = bh

where A is the area, b is the length of the base, and h is the height. We also know that the area of the parallelogram in this case is 25.2 cm² and the height is 4 cm.

Substituting these values into the formula, we get:

25.2 = b(4)

To solve for b, we can divide both sides by 4:

b = 25.2/4

b = 6.3

So the length of the base is 6.3 cm.

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The area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.

To find the area of a sector bounded by a 180° arc with a radius of 8 centimeters, you can follow these steps:

Step 1: Recall the formula for the area of a circle: A = πr², where A is the area and r is the radius.

Step 2: Calculate the area of the entire circle with a radius of 8 centimeters: A = π(8)² = 64π square centimeters.

Step 3: Determine the fraction of the circle represented by the 180° arc. Since a full circle is 360°, the fraction is 180°/360°, which simplifies to 1/2.

Step 4: Multiply the area of the entire circle by the fraction to find the area of the sector: (1/2) * (64π) = 32π square centimeters.

So, the area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.

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It takes about 1.5 seconds for the ball to hit the ground.

To find how long it takes for the ball to hit the ground, we need to find the value of x when h(x) = 0, since the height of the ball is 0 when it hits the ground. We can set -16x²+36 = 0 and solve for x:

-16x²+ 36 = 0

Dividing both sides by -16:

x² - 2.25 = 0

Adding 2.25 to both sides:

x²= 2.25

Taking the square root of both sides (we can ignore the negative root since time cannot be negative):

x = √(2.25) ≈ 1.5

Therefore, it takes about 1.5 seconds for the ball to hit the ground.

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The temperature are the times  indicated are:

T( 0) =   5.800°

T( 0.22) = -4.9094 °

T (0.44) = -2.6792 °

T(0.66) = 2.3244°

T(0.88)  = 4.8184°

T(1.1) = -4.1686 °

To solve the above, we need to use the formuala that we given which is T(x) = 5.8cos(3.8πx)

Entering the various values of x, can can obtain

T(0) = 5.8 cos (3.8π  (0) )  = 5.8cos(0) = 5.800

T(0.22) = 5.8 cos (3.8  π(0.22)) ≈ -4.9094

T(0.44) = 5.8cos (3.8π (0.44) ) ≈ -  2.6792

T(0.66) = 5.8cos(3.8π (0.66)) ≈ 2.3244

T(0.88) = 5.8cos(3.8π(0.88)) ≈ 4.8184

T(1.1) = 5.8 cos(3.8π(1.1  )) ≈ -4.1686

So the above answers are correct.

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The perimeter of the base of the triangular prism =

The area of the base of the triangular prism =

The total surface area of the triangular prism =

A triangular prism is a three-dimensional geometric shape that consists of two parallel triangular bases connected by three rectangular or parallelogram faces. The faces that connect the two bases are called lateral faces. The lateral edges are the edges that connect the lateral faces, and the base edges are the edges that form the triangles.

The perimeter of the base of the triangular prism

p = 10cm + 10cm + 12cm = 32cm

The area of the base of the triangular prism =

base area = (1/2) bh

= 1/2 × b × h = 1/2 × 12 × 8 = 48cm².

The total surface area of the triangular prism =  (Perimeter of the base × Length of the prism) + (2 × Base Area)

= (32 + 34) + (2 × 48) = 66 + 96 = 162cm².

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A. The partial derivatives are not equal, F is not conservative, potential function f(x, y) = N

B. The partial derivatives are equal, F is conservative, potential function f(x, y) = -3xy - [tex]x^2[/tex] + C

C. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

D. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

E. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

A. F(x, y) = (-6x – 7y) i +(-7x + 14y)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -6i - 7j

∂F/∂x = -7i + 14j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

B. F(x, y) = -3yi – 2xj

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3i

∂F/∂x = -2j

Since the partial derivatives are equal, F is conservative.

Potential function f(x, y) =[tex]-3xy - x^2 + C[/tex], where C is a constant.

C. F(x, y, z) = -3xi – 2yj+k

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -2j

∂F/∂z = k

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

D. F(x, y) = (-3 sin y)i + (-14y – 3x cosy)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3cosy j - 14i

∂F/∂x = -3cosy j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

E. [tex]F(x, y, z) = -3xi - 7yj + 7z^2k[/tex]

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -7j

∂F/∂z = 14zk

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

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Answer:

Modise is right.

This diagram does not represent a function because each input value does not correspond to exactly one output value. The input value 5 corresponds to two outputs, 2 and 9.

The function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

To determine if the function f(x) = –2x^3 + 39x^2 -216x + 6 has a local minimum or maximum, we need to find the critical points of the function and then determine the nature of those critical points.

First, we take the derivative of the function to find the critical points:

f(x) = –2x^3 + 39x^2 -216x + 6

f'(x) = –6x^2 + 78x - 216

f'(x) = –6(x^2 - 13x + 36)

f'(x) = –6(x - 4)(x - 9)

Setting f'(x) = 0, we get:

–6(x - 4)(x - 9) = 0

This gives us two critical points at x = 4 and x = 9.

To determine the nature of these critical points, we need to look at the sign of the derivative on either side of each critical point.

When x < 4, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

When 4 < x < 9, we have:

f'(x) = –6(x^2 - 13x + 36) > 0

When x > 9, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

This means that f(x) is decreasing on the interval (–∞, 4), increasing on the interval (4, 9), and decreasing on the interval (9, ∞). Therefore, we have a local minimum at x = 4 and a local maximum at x = 9.

To confirm this, we can evaluate the function at these critical points:

f(4) = –2(4)^3 + 39(4)^2 -216(4) + 6 = –26

f(9) = –2(9)^3 + 39(9)^2 -216(9) + 6 = 603

Therefore, the function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

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x = -5, dx/dt is equal to -2/25.

To find dx/dt, we need to use the chain rule of differentiation.

We know that dy/dt = -4 and we have the equation y = -5x^2 + 2.

Taking the derivative of both sides with respect to t, we get:

dy/dt = d/dt (-5x^2 + 2)

Using the chain rule, we can write this as:

dy/dt = (-10x) (dx/dt)

Now, we can plug in x = -5 and dy/dt = -4:

-4 = (-10(-5)) (dx/dt)

Simplifying, we get:

-4 = 50 (dx/dt)

Dividing both sides by 50, we get:

dx/dt = -4/50

Simplifying further, we get:

dx/dt = -2/25

Therefore, at x = -5, dx/dt is equal to -2/25.

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