An equivalent fraction for 6/8 is..?

The proof of "if a | b and b | a, where a and b are integers, then either a = b or a = -b" is shown below .

The representation "a|b and b|a ", means that "a" is a divisor of "b" and "b" is a divisor of "a".

which means , there exists an integer "c" ; such that b=ac    ...equation(1) and an integer "d" such that a = bd    .....equation(2)

Substituting the first equation into the equation(2) , we get:

⇒ a = bd = (ac)d = adc ;

Since a|a, it must be that either a = adc or adc = -a.

But  "c" and "d" are integers, So , "adc" must also be an integer.

Hence, it must be that adc = a.

Thus , a = adc = a. This implies that d = c = 1, and

So ⇒ a = b .....Equation(3) .

Similarly , if adc = -a, then we have: a = -a

Which implies that a = 0, but since "a" and "b" are integers, they cannot both be 0. So , it must be that a = b   ...equation(4) .

Therefore , On combining equation(3) and equation(4)  , we proved that "if a|b and b|a, where a and b are integers, then either a = b or a = -b" .

The given question is incomplete , the complete question is

Prove/show that if a|b and b|a, where a and b are integers, then either a = b or a = -b.

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You are 18 years old this year.

Let's call your age this year "x". Next year, your age will be "x + 1".

We know that the sum of the digits of "x" is three times bigger than the sum of the digits of "x + 1". Let's call the sum of the digits of "x" "s1", and the sum of the digits of "x + 1" "s2".

So we have the equation:

s1 = 3 * s2

If we add the digits of "x + 1", we get "s2 = a + b", where "a" and "b" are the digits of "x + 1".

Expanding the equation and substituting "s2":

s1 = 3 * (a + b)

Expanding the left side of the equation:

s1 = 3a + 3b

So the sum of the digits of "x" is three times the sum of the digits of "x + 1".

Now we need to find the values of "x" for which "s1" is three times bigger than "s2".

One possible solution is "x = 18", because:

So the answer is that you are 18 years old this year.

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You are 18 years old this year.

Let's call your age this year "x". Next year, your age will be "x + 1".

We know that the sum of the digits of "x" is three times bigger than the sum of the digits of "x + 1". Let's call the sum of the digits of "x" "s1", and the sum of the digits of "x + 1" "s2".

So we have the equation:

s1 = 3 * s2

If we add the digits of "x + 1", we get "s2 = a + b", where "a" and "b" are the digits of "x + 1".

Expanding the equation and substituting "s2":

s1 = 3 * (a + b)

Expanding the left side of the equation:

s1 = 3a + 3b

So the sum of the digits of "x" is three times the sum of the digits of "x + 1".

Now we need to find the values of "x" for which "s1" is three times bigger than "s2".

One possible solution is "x = 18", because:

"s1" = 1 + 8 = 9

"s2" = 1 + 9 = 10

9 = 3 * 10

So the answer is that you are 18 years old this year.

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The equation for the line that goes through (-10, 6) is given as y = -4x - 34

A parallel line has the same slope as the original line, but a different y-intercept. So, we want to find an equation of the form y = mx + b where m = -4 (the slope of the original line) and b is the y-intercept that gives the desired point (-10, 6).

We can find b by plugging in the coordinates of the point (-10, 6) into the equation y = mx + b:

6 = -4 * (-10) + b

6 = 40 + b

Subtracting 40 from both sides, we get:

-34 = b

So, the equation for the parallel line that goes through (-10, 6) is:

y = -4x - 34

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The value of the bulldozer after 12 years is 28,950.

Since it is given that the company depreciates the bulldozer linearly over its useful life,

The value of the bulldozer, as a function of time, should be in the form; y=m(t)+c.

where,

y= value of bulldozer

m and c = are some constants

And the boundary conditions are given as,

when t=0, y=78,750

= m(0)+c=78,750

= c = 78,750

and also given,

when t=16, y=12,350

=m(16 )+78,750=12,350

=m=12,350-78,750/16

=m=-4,150

so, the equation comes as,

y=78,750 - 4,150(t)

Value of bulldozer after 12 years would be,

y=78,750 - 4,150(12)

y=28,950

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Answer: 43

Step-by-step explanation: Each of the numbers is being added by 3.

4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52..

The initial amount in  Lea's account is 1212.12. Option A is the correct option.

What is the principal amount?

The amount you owe at any given time is the principal amount. When you first took out the loan, it is exactly your loan amount. The principal amount drops as you make EMI payments, and when it reaches zero, your loan is closed.

Given that  Lea earned $800 in simple interest on her investment after 11 years.

The formula of simple interest is

I = Prt.

P = Principal, r = rate of interest, t = time.

The rate of interest is r = 6% = 0.06, interest I = 800, time = 1 years.

Now putting the value of I, r,t in I = Prt:

800 = P×0.06×11

800= 0.66 P

Divide both sides by 0.66:

P = 800/0.66

P = 1212.1212

P = 1212.12(approx)

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We get to the conclusion that the response is 10x after examining the two sides.

The above equation's roots are 5 + 3i and 5 - 3i.

This indicates that x - (5+3i) and x - are the two factors of the provided polynomial (5 - 3i). The original statement must be the sum of the elements. We can thus write:

x² - (answer) + 34 = (x - 5 -3i)(x - 5 + 3i).........(1)

Simplifying the right hand side:

[tex]& (x-5-3 i)(x-5+3 i) \\[/tex]

[tex]& =x^2-5 x+3 x i-5 x+25-15 i-3 x i+15 i-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9(-1) \\[/tex]

[tex]& =x^2-10 x+25+9 \\[/tex]

[tex]& =x^2-10 x+34[/tex]

Thus, we can write the Equation 1 as:

[tex]x^2-\left(\text { answer) }+34=x^2-10 x+34\right.[/tex]

Our analysis of the two sides leads us to the conclusion that the response is 10x.

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Answer: slope = 7

Step-by-step explanation:

A graph of this system of equations on a coordinate plane is shown in the image attached below.

In order to to graph the solution to the given system of equations on a coordinate plane, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection;

y = 2x/5 + 4     ......equation 1.

-2x + 5y = -30     ......equation 2.

Making y the subject of formula in equation 2, we have the following:

5y = 2x - 30

Dividing both sides of the equation by 5, we have:

y = 2x/5 - 30

Based on the graph (see attachment), we can logically deduce that the solution to the given system of equations is non existent because the lines are parallel.

In conclusion, the given system of equations has no solution.

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Answer: 1/6 of a pound

Step-by-step explanation: Hello!

You know that Miss Collier bought a block of cheddar cheese that weighs 2/3 of a pound and she split it into four equal pieces.  This can be represented by the equation below:

2/3 divided by 4 = weight of a single slice of cheese

2/3 divided by 4/1 = weight of a single slice of cheese

Take the reciprocal of 4/1.  4/1 turns into 1/4.  Then multiply the two fractions.

2/3 x 1/4 = weight of a single slice of cheese.  

Multiply the numerators and denominators.  2 x 1 = 2, 3 x 4 = 12.

2/12 = weight of a single slice of cheese.

You can simplify this by dividing the numerator and denominator by 2.  2/2 = 1, 12/2 = 6.

The final answer is: 1/6 of a pound.

The solution is Option B.

The system of inequalities is x² - 2y ≥ 7000 and 2x + 5y < 1000

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the first inequality equation be represented as A

Let the second inequality equation be represented as B

Now , value of a company’s stock is represented by the expression x² - 2y

And , company’s goal is to maintain a stock value of at least $ 7,000

So , the inequality is x² - 2y ≥ 7000

And , the company’s purchases are modeled by 2x + 5y

Also , the company's purchases should be below $ 1,000

So , the inequality is 2x + 5y < 1000

Hence , the inequalities are solved

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Under the unbiased expectations theory of the term structure of interest rates, the expected one-year interest rate one year from now, E(2r1), is 10%.

The unbiased expectations theory states that the expected future rate (E(2r1)) is equal to the current rate (1R2). Therefore, the expected one-year interest rate one year from now, E(2r1), is 10%.

The unbiased expectations theory is a principle in finance that states that the expected future rate of return (E(2r1)) is equal to the current rate of return (r2). This theory suggests that market participants have rational expectations and their expectations are reflected in the current market prices. In other words, the theory posits that the current market prices incorporate all available information and reflect the best prediction of future prices.

In the example given, if the expected one-year interest rate one year from now is 10%, it means that market participants believe that the one-year interest rate in the future will be 10%. This expectation is reflected in the current market prices and is incorporated into the current rate of return.

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Answer: 14 - [tex]5x^{2}[/tex]

Step-by-step explanation:

So what I would do is remove those parentheses so you would have

8 + [tex]6x^{2}[/tex] + 6 - [tex]x^{2}[/tex]

Then you are going to combine like terms that would look like

14 - [tex]5x^{2}[/tex]

Hope that is correct

Answer: 4.5 pounds of flour equals to 72 ounces of flour. There are 14 days in 2 weeks, so 14 times 72 equals 1008 ounces of flour. If the baker uses all their 72 ounces of flour in one day to make loaves of bread, they will make 6 loaves of bread. That is also 84 loaves of bread if they make loaves of bread for 2 weeks straight.

Step-by-step explanation:

Answer:

Step-by-step explanation: Well, one lbs (pound) is equal to 16 oz (ounces)

So take the 4.5 lbs and multiply it by the 16 oz. That'll give you 224 oz

Then take the 224 oz and multiply that by 14, which is the number of days from two weeks. 224 multiplied by 14 is 3,136

To find the loaves of bread, take the 3,136 and divide it by 12, which gives you 261.3, or 261 1/3

The function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches is f(r) = 12π(r² + 3r + 3).

In this question, we are asked to determine a function for a spherical balloon that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches.

Volume of spherical balloon = V(r) = 4/3 π r³

If the radius of balloon is r+3 inches = V(r+3) = 4/3 π (r+3)³

The amount of air required is the difference between V(r) and V(r+3):

So, f(r) = V(r+3) - V(r) = 4/3 π (r+3)³ - 4/3 π (r)³

                                  = 4/3 π ((r+3)³ -r³)

                                  = 4/3 π (r+3 -r)((r+3)² + r(r+3) + r²)

                                  = 4π(r² + 9 + 6r + r²+3r + r²)

                                  = 4π(3r² + 9r + 9)

                             f(r) = 12π(r² + 3r + 3)

Hence, the function representing is f(r) = 12π(r² + 3r + 3).

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The minimum amount of number of divisions made by Euclid's algorithm among all inputs 1 to 10 is 0, as the greatest common divisor of any two numbers between 1 and 10 must be 1. The maximum number of divisions made by Euclid's algorithm among all inputs 1 to 10 is 19, as the greatest common divisor of 8 and 10 requires 19 divisions.

Euclid's algorithm is an efficient method for finding the greatest common divisor (GCD) of two integers. It works by repeatedly dividing larger numbers by smaller numbers until the remainder is 0. The GCD of two integers is then equal to the smaller number. The minimum number of divisions made by Euclid's algorithm among all inputs 1 to 10 is 0, as the GCD of any two numbers between 1 and 10 must be 1. This is because any two numbers between 1 and 10 have only 1 as their common divisor. On the other hand, the maximum number of divisions made by Euclid's algorithm among all inputs 1 to 10 is 19, as the GCD of 8 and 10 requires 19 divisions. This is due to the fact that 8 and 10 have no common divisors except for 1, so the algorithm must divide 8 by 10 repeatedly until the remainder is 0. Thus, the maximum number of divisions made by Euclid's algorithm among all inputs 1 to 10 is 19.

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The value of x is 44.

The most basic type of triangle is an equilateral triangle, which has three sides of equal length and three equal angles. Another common type of triangle is the right triangle, which has one right angle (90 degrees) and two other angles that add up to 90 degrees. Scalene triangles have no sides or angles of equal length, while isosceles triangles have at least two sides of equal length.

Triangles have many interesting and useful properties that can be used in geometry and other areas of mathematics. For example, the Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side. This theorem can be used to find the length of a missing side in a right triangle.

Triangles also have an important relationship with circles. The circumcenter of a triangle is the center of the circle that passes through all three vertices of the triangle, while the incenter of a triangle is the center of the circle that is tangent to each of the triangle's sides.

From the triangle angle sum theorem, we know that the sum of all angles in a triangle is 180 degrees. So for triangle ABC, we have:

angle ABC + angle BCA + angle CAD = 180

50 + (3x - 10) + 2x = 180

Combining like terms and solving for x, we get:

5x = 220

Therefore, x = 44.

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The scale factor is equal to 4 which is calculated by determining the coordinates.

Let's begin by determining the coordinates.

These are the ones we begin with!

D (1,1)

E (2,2)

F (4,2)

G (3,1)

Now we must locate the best image cords.

D' (4,4)

E' (8,8)

F' (16,8)

G' (12,4)

We're looking for the scale factor, so we'll see what number is multiplied by those coordinates to get the prime coordinates.

D is (1,1) and D' is (4,4), but we know that 1x4 equals 4, so 4 is the scale factor by which we multiply all of the numbers.

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A sequence is characterised as one that adheres to a predetermined pattern. The following term is created by increasing or decreasing the previous word by a certain amount.

On occasion, an expression is followed by every term in the series. The general formula for an AP is T n = a + (n - 1) d.

Understanding the underlying pattern or rule that creates the sequence is necessary for formulating the general term an of the sequence. Finding a formula for the overall term of a series can be done in a number of ways, including:

1) Finding a pattern: In some cases, a pattern can be found by examining the terms of the sequence and looking for a common ratio or difference. For instance, the formula for the general term can be written as a = a1 + (n - 1)d, where a1 is the first term and d is the common difference, assuming the terms of a sequence are rising by a constant amount.

2) Recurrence relations are used to characterise some sequences. These relations define the nth term in terms of the phrases that came before it. If the recurrence relation, for instance, is a = an-1 + an-2, then the general term's formula can be discovered by resolving the recurrence relation.

3) Making use of generating functions: Sequences can be represented mathematically using generating functions. A formula for the general term of a sequence can be discovered by fiddling with the generating function.

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The K-Means algorithm terminates when. a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

The K-Means algorithm terminates when

A) a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

B) the cluster centers for the current iteration are identical to the cluster centers for the previous iteration.

C) the number of instances in each cluster for the current iteration is identical to the number of instances in each cluster of the previous iteration.

D) the number of clusters formed for the current iteration is identical to the number of clusters formed in the previous iteration.

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The preparation of Company X's balance sheet as the current date is as follows:

Company X

As of current date

Cash                                 $8,250

Inventory                            1,225

Accounts Receivable          675  $10,150

Long-term investment  $1,000

Land and building          8,878

Equipment                      4,230    $14,108

Total Assets                               $24,258

Accounts Payable         $485

Short-term loan               500       $985

Long-term loan                         $9,982

Total liabilities                         $10,967

Initial capital           $12,000

Retained earnings       1,291  $13,291

Total liabilities and equity   $24,258

The balance sheet is a financial statement that shows the assets, liabilities, and equity balances of an entity at a point in time.

Assets describe the things owned by the entity while liabilities refer to its debts.  The difference between assets and liabilities is the equity (or the portion of assets belonging to the owners).

Initial investment = $12,000

Retained Earnings = $1,291

Long-term loan = $9,982 ($12,000 - $2,018)

Land and building = $8,878

Equipment = $4,230

Cash $8,250

Inventory = $1,225

Accounts Receivable = $675

Accounts Payable = $485

Short-term loan = $500

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Prepare Company X's balance sheet.

47 degrees is the the angle measure of one of the base angles.

A triangle is a three-sided polygon that consists of three edges and three vertices.

We know that an isosceles triangle is composed of a vertex angle and two base angles.

The base angles are congruent to each other.

We also know that the sum of a triangle's angles is 180 degrees.

86+2x=180

Subtract 86 from both sides

2x=180-86

2x=94

Divide both sides by 2

x=47

Hence, 47 degrees is the the angle measure of one of the base angles.

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The value of k is "i".

What is cube and square?

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

a^3 + A^2 = 0

Substituting A = -1 into this equation gives us:

a^3 + (-1)^2 = 0

Simplifying the right-hand side of this equation:

a^3 + 1 = 0

So,

a^3 = -1

Taking the cube root of both sides:

a = -1^(1/3)

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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The complex number -1, which is represented as an imaginary number, or I has the value of k, which is the cube root of that number.

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

[tex]a^3+A^2=0[/tex]

Substituting A = -1 into this equation gives us:

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

Simplifying the right-hand side of this equation:

[tex]a^3+1=0[/tex]

So,

[tex]a^=-1[/tex]

Taking the cube root of both sides:

[tex]a=(-1)^{\frac{1}{3} }[/tex]

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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8/4=12/x is the proportions could be used to find the width, w, of his model.

Completed question :

grace is making a scale model of her favorite car. the actual car is 8 feet long and 4 feet wide. grace wants her model to be 12 inches in length. which proportion could be used to find thethe width, w, of his model?

A:8/4=12/w B:8/4=w/12 C: 8/12=w/4 D:12/8=4/w

Given data :

length = 8 feet

width = 4 feet

Length that grace wants = 12 feet

8/4=12/x

8feet long           12 inches

4feet wide           x = width which is not given

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Answer: 1 7/10

Step-by-step explanation:

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

A circle's chord is the line segment that connects any two locations on the circle's circumference. It should be noted that a circle's diameter is defined as the lengthiest chord that runs through the centre of the circle.

KL and MN are chords of circle Q. They intersect at point J .MKN = (10x + 2 )° , mLM = (3x–4)°, and m∠KJM=103°.

(10x + 2 )° +  (3x–4)° +103° = 180 degree

13x + 2 + 103 = 180 degree

13x = 180 - 103 = 77

x = 77 / 13 = 5.92 .

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

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a = ( 0 , - 4 )

b = ( - 2 , - 2 )

[tex]m = - \frac{y(a) - y(b)}{x(a) - x(b)} \\ [/tex]

[tex]m \: = \frac{ - 4 - ( - 2)}{0 - ( - 2)} \\ [/tex]

[tex]m = \frac{ - 4 + 2}{2} \\ [/tex]

[tex]m = \frac{ - 2}{2} \\ [/tex]

[tex]m = - 1[/tex]

A) f(x) +5

Domain: x ∈ (5, 15)

Range: f(x) ∈ (12, 22)

B)  f(x + 5)

Domain: x ∈ (0, 10)

Range: f(x) ∈ (7, 17)

C) f(5x)

Domain: x ∈ (1, 3)

Range: f(x) ∈ (7, 17)

D) 5f(x)

Domain: x ∈ (5, 15)

Range: f(x) ∈ (35, 85)

Now, According to the question:

Domain:

Consider a function y = t(x), where y is the dependent variable and x is the independent variable. Domain value represents the value(s) of the independent variable at which the given function is defined. On the other hand, the corresponding values of y represents the range of that function.

f(x) has a domain of (5, 15) and a range of (7, 17)

A) f(x) + 5

The domain will be the same as the domain of f(x) but the range will be (7 + 5 , 17 + 5) i.e., (12, 22)

Domain: x ∈ (5, 15)

Range: f(x) ∈ (12, 22)

B) f(x + 5)

The domain will be (5 -5, 15-5) i.e., (0, 10) but the range will be the same.

Domain: x ∈ (0, 10)

Range: f(x) ∈ (7, 17)

C) f(5x)

The domain will be [tex](\frac{5}{5},\frac{15}{5} )[/tex] i.e., (1, 3) but the range will be the same.

Domain: x ∈ (1, 3)

Range: f(x) ∈ (7, 17)

D) The domain will be the same as the domain of f(x) but the same will be (5× 7, 17× 5) i.e., (35, 85)

Domain: x ∈ (5, 15)

Range: f(x) ∈ (35, 85)

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The given question is incomplete,

So, This question like that the given question:

Suppose that f(x) has a domain of (5, 15) and a range of (7,17).

What are the domains and the ranges of the following?

A) f(x) +5

B) f(x + 5)

C) f(5x)

D) 5f(x)

The slope of the tangent line to the graph of f(x) = −x² + 4x at x = 4 is 0, and the equation of the tangent line is y = 0.

The slope of a tangent line to a graph of a function is a measure of how steeply the function is increasing or decreasing at a certain point. A tangent line is a line that just touches the graph at a single point and has the same slope as the graph at that point.

Here we need to find the slope of the tangent line to the graph of

=> f(x) = −x² + 4x at x = 4,

we first need to determine the derivative of the function. The derivative of the function gives us the rate of change of the function, which is the slope of the tangent line.

Here The derivative of f(x) = −x² + 4x is given by

=> f'(x) = −2x + 4.

Now we have to find the slope of the tangent line at x = 4, we evaluate the derivative at x = 4:

=> f'(4) = −2 * 4 + 4 = 0.

So, the slope of the tangent line at x = 4 is 0.

To know more about tangent line here.

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