Please help with number 3!! Thank you so much!

The area of the shaded region is,

(3) 100.28 sq. unit

(4) 72 sq.m

(5) 49.09  sq.m

What is the area?

The area of an object is the amount of room it occupies in two dimensions. It is the calculation of how many unit squares entirely encircle the area of a closed figure.

The accepted measure of area is the square unit, which is frequently expressed as square inches, square feet, etc. Most shapes and objects have edges and angles.

(3)

To calculate the area of the semi-circle having a radius of 3 cm,

[tex]A_1=\frac{1}{2} \pi r^2\\A_1=\frac{1}{2} \pi(3)^2\\A_1=14.14 \ \ sq. \ cm[/tex]

To calculate the area of the rectangle having a width of 12 cm and a height of 6 cm,

[tex]A_2=l*w\\A_2=12*6\\A_2=72 \ sq.cm[/tex]

To calculate the area of the semi-circle having a radius of 3 cm,

[tex]A_3=\frac{1}{2} \pi (3)^2\\A_3=14.14[/tex]

The shaded region's size was calculated using

[tex]A=A_1+A_2+A_3\\A=14.14+72+14.14\\A=100.28 \ \ sq.cm[/tex]

(4)

To calculate the area of the rectangle having a width of 13 m and a height of 8 m,

[tex]A_1=l*w\\A_1=13*8\\A_1=104 \ sq.m[/tex]

To calculate the area of an unshaded triangle,

[tex]A_2=\frac{1}{2} *b*h\\A_2=\frac{1}{2} *8*8\\\\A_2=32 \ sq.m[/tex]

The shaded region's size is thus,

[tex]A=A_1-A_2\\A=104-32\\A=72 \ sq.m[/tex]

(5)

The shaded region's size was calculated using

[tex]A=Area \ of \ circle-Area \ of \ square\\A=\pi (6)^2-(8)^2\\A=36\pi -64\\A=49.09 \ sq.m[/tex]

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The 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners is (62.49,69.51). So, the Lower and Upper end point are 62.49, 69.51 respectively.

We have a survey of 53 randomly selected homeowners. So,

Sample size, n = 53

Mean spend of home maintenance, μ

= $66 per month

Standard deviations, σ = $ 11 per month

We have to determine the Lower and Upper end point or bound of confidence interval. First we calculate the Z score for 98% of confidence interval. The level of significance, α = 0.02 and α/2 = 0.01, so using distribution table [tex]z_{ \frac{\alpha}{2}} = 2.326[/tex]

Now, Margin of error , [tex]MOE = z_{\frac{\alpha}{2}} ( \frac{\sigma }{\sqrt{}n}) [/tex]

[tex] = 2.326 ( \frac{11}{ \sqrt{}53 })[/tex]

=> E = 3.51

At 98% confidence interval estimate of the population mean is, [tex]\bar x - E < \mu < \bar x + E[/tex]

[tex]66 - 3.51 < \mu < 66 + 3.51[/tex]

=> 62.49 < μ < 69.51

Therefore, Lower end point or bound

= 62.49 and Upper end point or bound

= 69.51.

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Answer:

165.5

Step-by-step explanation:

Because you have to add them together and then divide

Answer:

43. All real numbers

44. x = 1/4

45. No solution

Step-by-step explanation:

The technique here is two write two expressions with the same base that are equal. Then set the exponents equal and solve for the variable.

43.

[tex] 3^{3x + 6} = 27^{x + 2} [/tex]

[tex] 3^{3x + 6} = (3^3)^{x + 2} [/tex]

[tex] 3^{3x + 6} = 3^{3(x + 2)} [/tex]

[tex] 3^{3x + 6} = 3^{3x + 6} [/tex]

[tex] 3x + 6 = 3x + 6 [/tex]

Answer: all real numbers

44.

[tex] 3^{4x + 3} = 81 [/tex]

[tex] 3^{4x + 3} = 3^4 [/tex]

[tex] 4x + 3 = 4 [/tex]

[tex] 4x = 1 [/tex]

[tex]x = \dfrac{1}{4}[/tex]

45.

[tex] 4^{x + 3} = 2^{2(x + 1)} [/tex]

[tex] (2^2)^{x + 3} = 2^{2(x + 1)} [/tex]

[tex] 2^{2(x + 3)} = 2^{2(x + 1)} [/tex]

[tex] 2(x + 3) = 2(x + 1) [/tex]

[tex] x + 3 = x + 1 [/tex]

[tex] 3 = 1 [/tex]

No solution.

Answer: 4

Step-by-step explanation:  8x11=88.

352=88x

x=352/88

x=4

Check:

11x8x4=352

Answer:

$35.75

Step-by-step explanation:

$6.50 + $3.90(7.5) = $6.50 + $29.25 = $35.75

Therefore, her net income is $77408.

A coefficient is a numerical factor that is multiplied by a variable or a term in an algebraic expression. In mathematics, coefficients are commonly used in polynomial functions, where they determine the degree of the polynomial and the specific values of the function.

For example, in the polynomial function[tex]f(x) = 3x^2 + 2x - 1[/tex], the coefficients are 3, 2, and -1. The coefficient 3 is multiplied by the variable. [tex]x^2[/tex], the coefficient 2 is multiplied by the variable x, and the coefficient -1 is the constant term.

Coefficients are also commonly used in statistics, where they are used to describe the relationship between variables in a statistical model. In regression analysis, for example, coefficients are used to represent the slope and intercept of a line that best fits the data.

by the question.

To calculate her net income, we need to subtract the total tax paid from her gross income:

[tex]Net Income = Gross Income - Income Tax - Medicare Levy[/tex]

Substituting the given values, we get:

[tex]Net Income = $105500 - $26982 - $2110[/tex]

[tex]Net Income = $77408[/tex]

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The inequality that has the solutions represented by the graph?

2x - 5 > 3

The solution of the inequality represented by the graph is x > 4

The solution of the given inequalities are:

5 - x > 4;

solution: x < 1

2x + 1 ≥ 9;

solution: x ≥ 4

2x - 5 > 3

solution: x > 4

6x - 7 > 5

solution: x > 3

2. Considering the inequality statement given:

n ≤ 3: This is a solution.

If we substitute n with 3, we get 5 + 2(3) ≤ 11, which simplifies to 11 ≤ 11, which is true. And if we substitute n with any number less than 3, the inequality will still hold.

3 ≥ n: This is also a solution.

If we substitute n with 3, we get 5 + 2(3) ≤ 11, which simplifies to 11 ≤ 11, which is true. And if we substitute n with any number greater than 3, the inequality will still hold.

n ≤ 8: This is a solution.

If we substitute n with 8, we get 5 + 2(8) ≤ 11, which simplifies to 21 ≤ 11, which is false. But if we substitute n with any number less than or equal to 8, the inequality will still hold.

3 ≥ n: This is the same as the second representation, so it is also a solution.

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Complete question:

Which inequality has solutions represented by the graph?

5 - x > 4

2x + 1 ≥ 9

2x - 5 > 3

6x - 7 > 5

Consider the inequality Five plus two times a number n is less than or equal to eleven. Indicate whether each representation is or is not a solution.

n ≤ 3

3 ≥ n

n ≤ 8

3 ≥ n

Answer:

Find the components of the line (x,y)

And then use Tan ^-1(y/x)= Theta or angle

I hope you understand, and I go explain further.

Step-by-step explanation: In "simple terms". Slope is rise/run. Rise is the change in y value, which will equal the length of the opposite side of the right triangle. Run is the change in x value, which is the length of the adjacent side. The trigonometric function Tangent is opposite/adjacent. Use the inverse tangent of the slope to find the angle.

The only catch is that sometimes inverse tangent will give you a negative angle. It is generally not acceptable to provide the angle in a triangle as the negative angle. If you get a negative angle, use the absolute value of the angle.

Answer:

9 x 17 is greater than 9 x 15

Step-by-step explanation:

You can tell without finding the product because 17 is greater than 15, and the higher the number you multiply by, the larger the solution will be. I hope this helped :)

Answer:

The answer is A: 2500 pi

Step-by-step explanation:

To find the volume of a cylinder use the formula:

π[tex]r^{2}h[/tex]

So we know that the diameter is 10, so radius must be 10/2 or 5.

Now we square 5 to get 25.

Multiply 25 times height which is 100.

So 25 x 100 is equal to 2500

Now multiply the pi or just let it stay like that

The answer is 2500π.

Hope this helps!

The answer is 8.6+(-6.1). This is because addition and subtraction are inverse operations.

Subtraction that involves taking one number away from another. It is the inverse of addition and is represented by the minus sign (-).

This means that when you add two numbers and then subtract the same amount, the result is the same as just adding the two numbers.

Similarly, when you subtract two numbers and then add the same amount, the result is the same as just subtracting the two numbers.

Therefore, 6.1+8.6 is equivalent to 8.6+(-6.1).

Using mathematical notation, we can represent this expression as 6.1+8.6 = 8.6+(-6.1).

To add and subtract the numbers, we can use the traditional column method.

First, we write out 6.1+8.6 as 6.1 - (-8.6). Then, we add the numbers in each column, starting with the ones column:

Ones column: 1+6=7

Tens column: 8-8=0

And our answer is 7,0. We can also write this as 8.6+(-6.1) to show that the two expressions are equivalent.

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[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{\sqrt{257}}\\ a=\stackrel{adjacent}{x}\\ o=\stackrel{opposite}{15} \end{cases} \\\\\\ (\sqrt{257})^2= (x)^2 + (15)^2\implies 257=x^2 + 225 \\\\\\ 32=x^2\implies \sqrt{32}=x\implies 4\sqrt{2}=x\implies 5.66\approx x[/tex]

Let's assume that the width of the rectangular poster is "x".

According to the problem, the length is 8 more inches than half the width, which can be expressed as:

length = (1/2)x + 8

The area of the poster is given as 40 square inches, so we can write the equation:

Area = length × width

Substituting the values of length and width in terms of "x", we get:

40 = ((1/2)x + 8) × x

Simplifying the equation:

40 = (x^2)/2 + 8x

Multiplying both sides by 2 to eliminate the fraction:

80 = x^2 + 16x

Rearranging the equation in standard quadratic form:

x^2 + 16x - 80 = 0

Using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = 16, and c = -80

x = (-16 ± sqrt(16^2 - 4(1)(-80))) / 2(1)

x = (-16 ± sqrt(736)) / 2

x = (-16 ± 8sqrt(9)) / 2

We can discard the negative solution since width cannot be negative:

x = (-16 + 24) / 2

x = 4

So, the width of the poster is 4 inches.

Using the equation for length:

length = (1/2)x + 8

length = (1/2)(4) + 8

length = 2 + 8

length = 10

Therefore, the dimensions of the poster are 10 inches by 4 inches.

The values of Mean = 6.3 and Median = 6

A dot plot is a type of data visualization that displays data points as dots on a number line. Each dot represents a single data point and is placed at the corresponding value on the number line.

Here, each dot corresponds to a particular value in the data set, below diagram.

Mean: The mean is calculated by dividing the total value of all the data sets by the total number of data sets.

You can figure out the total as follows:

0 (1) = 0            4 (3) = 12             5(8) = 40          6(3) = 18

7(1) = 7              8(5) = 40             9(2) = 18           10 (3) = 30

Sum = 0+12+40+18+7+40+18+30 = 165

Number of data set = 26

⇒Mean = 165/26

             = 6.346 ≈ 6.3               (nearest tenth place)

⇒Median: The center value in the data collection is known as the median. Since there are 26 data points total, the middle value is located between data points 13 and 14. The median value will be determined by averaging the 13th and 14th data points.

So, the 13th and 14th values both are 6.

Therefore, median = {6+6} ÷ 2 = 12/2 = 6

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Complete Question-

a) The cost at production level 1200 is not given in the text, b) The average cost at production level 1200 is 2101.75, c) The marginal cost at production level 1200 is 3200, d) The production level that minimizes the average cost is not meaningful as it is outside the domain of production levels, e) The minimal average cost is not meaningful as it cannot be negative.

An expression is a mathematical phrase that can contain variables, constants, and operators (such as addition, subtraction, multiplication, division, and exponentiation) that represent some combination of numbers or other mathematical objects. Expressions can be simple, such as a single number or variable, or complex, such as a series of terms connected by operators. Expressions can be evaluated, simplified, and manipulated using algebraic techniques to solve problems and analyze mathematical relationships.

a) The cost at the production level 1200 is[tex]C(1200) = 12100 + 800(1200) + (1200)^2 = 2522100.[/tex]

b) The average cost at the production level 1200 is [C(1200)]/1200 = 2101.75.

c) The marginal cost at the production level 1200 is the derivative of C(x) with respect to x evaluated at x = 1200. Thus, MC(1200) = 800 + 2(1200) = 3200.

d) The production level that will minimize the average cost is given by the formula x = -b/(2a) where a and b are the coefficients of [tex]x^2[/tex] and x, respectively. Thus, x = -800/(2(1)) = -400. This value is not meaningful, as it is not in the domain of production levels.

e) The minimal average cost is the average cost at the vertex of the parabola given by C(x). The x-coordinate of the vertex is x = -b/(2a) = -800/(2(1)) = -400, and the y-coordinate is C[tex](-b/(2a)) = C(-400) = 12100 + 800(-400) + (-400)^2 = 168900[/tex]. Therefore, the minimal average cost is 168900/(-400) = -422.25. This is not a meaningful answer, as average cost cannot be negative.

Therefore,a) The cost at production level 1200 is not given in the text, b) The average cost at production level 1200 is 2101.75, c) The marginal cost at production level 1200 is 3200, d) The production level that minimizes the average cost is not meaningful as it is outside the domain of production levels, e) The minimal average cost is not meaningful as it cannot be negative.

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a) A 2*2 area model would not work to multiply (x + 3)(2x⁴ − 3x³ − 2x² − 4x − 1) because this expression has 5 terms, and a 2*2 area model can only accommodate 4 smaller areas.

b) The area model for (x + 3)(2x⁴ − 3x³ − 2x² − 4x − 1) with the correct lengths and widths labeled has been attached below.

c) The final total area is 2x² - 10x - 15.

An area model is a way to visualize multiplication by breaking up the larger rectangle into smaller rectangles whose areas represent the products of the terms being multiplied. To determine the correct rows and columns for the area model, we count the number of terms in each factor and use that as a guide. For example, if one factor has 3 terms and the other has 4 terms, we would use a 3x⁴ area model, with 3 rows and 4 columns.

a) A 2*2 area model would not work to multiply (x + 3)(2x⁴ − 3x³ − 2x² − 4x − 1) because this expression has 5 terms, and a 2*2 area model can only accommodate 4 smaller areas.

b) The area model for (x + 3)(2x⁴ − 3x³ − 2x² − 4x − 1) with the correct lengths and widths labeled has been attached below.

    2x⁴ − 3x³ − 2x² − 4x − 1

 +---------------------------------------

x  |   2x⁵    -3x⁴    -2x³    -4x²   -x

 |

3  |   6x⁴    -9x³    -6x²    -12x    -3

 +---------------------------------------

c) To find the total area of a rectangle with length (x + 8) and width (x − 5), we can use an area model with 2 rows and 2 columns. The length (x + 8) corresponds to one dimension of the rectangle, while the width (x − 5) corresponds to the other dimension. The area of each smaller rectangle in the model represents the product of one term from each factor.

     x + 8      x - 5

 +--------------------

x  |  x² + 8x   x² - 5x

 |

-5 | -5x + -40   -5x + 25

 +--------------------

The final total area can be found by adding up the areas of the smaller rectangles:

Total area = x² + 8x + x² - 5x - 5x - 40 + 25

= 2x² - 10x - 15

= 2x² - 10x - 15.

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Answer:

Step-by-step explanation:

An irrational number is a number that cannot be expressed as a simple fraction. Therefore 0.6263646 is irrational.

The formula for Py(Y) = ∑P y,z (y, z) for z ∈ Z is consistent with the third Axiom of the probability measure.

The third Axiom of the probability measure states that the sum of probabilities of all possible outcomes of an event is equal to 1. In the context of two discrete random variables Y and Z, this axiom means that the sum of probabilities of all possible pairs (y, z) is equal to 1.

The formula PY(Y) = ∑P y,z (y, z) for z ∈ Z is related to this axiom because it represents the probability of the random variable Y taking on a specific value y, regardless of the value of Z. The summation is taken over all possible values of z in Z, and P y,z (y, z) represents the joint probability of Y and Z taking on the specific values (y, z).

Since the sum of probabilities of all possible pairs (y, z) is equal to 1, the sum of probabilities of all possible values of Y, represented by PY(Y), must also be equal to 1.

Therefore, the formula for PY(Y) is consistent with the third Axiom of the probability measure.

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Answer:

There are 32 apples in the bag.

Step-by-step explanation:

To find the number of apples in the bag, we can use the formula:

number of apples = total mass of apples / average mass of each apple

Given that the bag of apples has a mass of 4 kg, and the average mass of each apple is 1/8 kg, we can substitute these values into the formula to get:

number of apples = 4 kg / (1/8) kg/apple

Simplifying the right-hand side by dividing 4 kg by (1/8) kg/apple, we get:

number of apples = 4 kg * 8/apple = 32 apples

Therefore, there are 32 apples in the bag.

Answer:

Step-by-step explanation:

It is simple by averaging formula as follows
Total mass/average mass = Total apples
4/(1/8)= 32 Apples

Answer:

a.) P(x)= 3x - 1120

b.) make $1,180 profit in 1 week by selling 800 calculators

Step-by-step explanation:

a.) P(x)=R(x)-C(x)

R(x)=11x; C(x)=8x + 1220   assuming they are asking for profit in a week

P(x)= 3x - 1220

b.) P(800) = 3(800) -1220 = 1180

For any rectangle ABCD the statement;

1. ∠A ≅ ∠D is always true

2. [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex] is sometimes true

A rectangle is a quadrilateral that has two pairs of parallel and congruent opposite sides and four right angles (90 degree angles). A rectangle is a special case of a parallelogram.

1. ∠A ≅ ∠D

The above statement is always true for rectangles. In Euclidean geometry, by the definition of a rectangle, the four interior angles are right angles, which means m∠A = m∠B = m∠C = m∠D = 90°, therefore; ∠A ≅ ∠B ≅ ∠C ≅ ∠D.

The statement; ∠A ≅ ∠D, according to the above expression and the transitive property of congruence, is always true.

Please find attached the drawing of the rectangle, created with MS Word that can be used to illustrate the congruence of the angles with the angle marks indicating perpendicular sides, forming 90° angles, which indicates that the angles are congruent.

   A_____[tex]{}[/tex]_____B

    |            [tex]{}[/tex]          |

    |           [tex]{}[/tex]           |

    |          [tex]{}[/tex]            |

    |          [tex]{}[/tex]            |

    |           [tex]{}[/tex]           |

    |____[tex]{}[/tex]______|

    D                     [tex]{}[/tex] C

2. [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex]

The above statement is sometimes true for rectangles. The definition of a rectangle indicates that the two pairs of opposite sides are congruent. In particular, [tex]\overline{AB}[/tex] and [tex]\overline{CD}[/tex] are congruent, aa well as sides [tex]\overline{BC}[/tex] and [tex]\overline{AD}[/tex] are also congruent. The sides [tex]\overline{BC}[/tex] and [tex]\overline{CD}[/tex] are congruent if or when the rectangle is a square, therefore, the statement [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex] is sometimes true.

Please find attached the drawings created with MS Word which illustrates when [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex] and when [tex]\overline{BC}[/tex] [tex]\ncong[/tex] [tex]\overline{CD}[/tex]

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Let us assume that the variables:

d = distance

t = time (in seconds)

~

The question assumes that the horse will move 15 meters every second. Therefore:

d = 15t

IF:

t (time) = 1 seconds, then:

d = 15(1)

d = 15

IF:

t (time) = 2 seconds, then:

d = 15(2)

d = 30

IF:

t (time) = 5 seconds, then:

d = 15(5)

d = 75

~

As you can see, the graphed numbers are much more larger than the numbers that you have already written, and so it is best to change the numbers. Generally the y-axis is used for time (in this case, seconds), whlie the x-axis is used for distance (in this case, meters).

~

The triangles on the front of the Louvre Pyramid are all isosceles triangles.

The building acts as a visible reminder of the importance of the museum's Egyptian Antiquities collections.

The primary entry point to the Louvre is the I.M. Pei Pyramid, which is situated in the courtyard. The building acts as a visible reminder of the importance of the museum's Egyptian Antiquities collections.

A graph on the right AB = 35 meters {Given}

AE = 32 meters {Given}

Because AE = BE = 32 meters

So <EAB = <ABE By measure,

<EAB=663°

So <ABE=663°.

So <AEB = 180° - LEAB-LABE.

= 180°-66.3° - 66.3° =47.4°

So, ΔABE is an isosceles triangle.

And the four triangles are all isosceles triangles.

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After answering the provided question, we can conclude that Therefore, circle the length of the minor arc AB is approximately 50.9 cm.

A circle appears to be an a double component that is defined as the collection of all points in a jet that are equidistant out from hub. A circle is typically depicted with a capital "O" for the centre and a bottom end "r" for the radius, which represents the distance from where it started to any point on the circle. The formula 2r gives the girth (the distance from the center of the circle), where (pi) seems to be a proportionality steady roughly equal to 3.14159. The formula r2 computes the circumference of a circle, which is the quantity of space inside the circle. To calculate the length of the minor arc AB, we need to first find the central angle that it subtends.

C = 2πr

r = C/(2π)

r = 65/(2π) ≈ 10.34 cm

angle = (arc length / radius) x (180/π)

ADB = 360° - AOB

ADB = 360° - 72° = 288°

angle ADB = (arc length AB / 10.34) x (180/π) = 288°

arc length AB = (288° x 10.34 cm x π) / 180 ≈ 50.9 cm

Therefore, the length of the minor arc AB is approximately 50.9 cm.

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The volume of the triangular prism is 135 cm³.

To calculate the volume of a triangular prism, we multiply the area of the triangular base by the height of the prism.

Given the dimensions provided:

Base: Length = 5 cm, Width = 9 cm

Height: 6 cm

To find the area of the triangular base, we use the formula for the area of a triangle:

Area = (1/2) * base * height

Substituting the values:

Area = (1/2) * 5 cm * 9 cm

Area = 22.5 cm²

Now, to calculate the volume of the prism, we multiply the area of the base by the height:

Volume = Area * Height

Volume = 22.5 cm² * 6 cm

Volume = 135 cm³

Consequently, the triangular prism has a 135 cm3 volume.

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There are 2,000 pounds in one ton. Therefore, to convert 25 tons to pounds, we can multiply 25 by 2,000:

25 tons x 2,000 pounds/ton = 50,000 pounds

So there are 50,000 pounds in 25 tons.

Answer:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33

Step-by-step explanation:

All of these work. I hope this helps! Pls give brainliest!

Answer:

Your answer would be  

x/3<12

(i) The domain of f(x) is all real numbers, since there are no restrictions on the input variable x.

(ii) We have:

f(x) = x^4 - 2x^3

f'(x) = 4x^3 - 6x^2

f''(x) = 12x^2 - 12x

(iii) To find the critical points, we need to solve the equation f'(x) = 0:

4x^3 - 6x^2 = 0

2x^2(2x - 3) = 0

x = 0 or x = 3/2

So the critical points are (0,0) and (3/2, -27/16).

(iv) To determine where f(x) is increasing or decreasing, we need to examine the sign of f'(x) on different intervals. We can make a sign chart for f'(x):

| x | -∞ | 0 | 3/2 | +∞ |

|---------|--------|-------|-------|--------|

| f'(x) | - | 0 | + | + |

From the sign chart, we see that f(x) is decreasing on the interval (-∞, 0) and increasing on the interval (0, 3/2) and (3/2, +∞).

(v) To find the relative extrema, we need to examine the sign of f'(x) around the critical points. We can make a table:

| x | 0- | 0+ | 3/2- | 3/2+ |

|---------|-------|-------|-------|-------|

| f'(x) | - | + | - | + |

| f(x) | 0 | 0 | -27/16| -27/16|

From this table, we see that f(x) has a relative minimum of -27/16 at x = 3/2, and no relative maximum or minimum at x = 0.

(vi) To find the intervals where f(x) is concave up or concave down, we need to examine the sign of f''(x) on different intervals.

We can make a sign chart for f''(x):

| x | -∞ | 0 | 1 | +∞ |

|---------|--------|-------|------|--------|

| f''(x) | + | - | + | + |

From the sign chart, we see that f(x) is concave down on the interval (-∞, 0) and concave up on the intervals (0, 3/2) and (3/2, +∞).

(vii) To find the points of inflection, we need to solve the equation f''(x) = 0:

12x^2 - 12x = 0

12x(x - 1) = 0

x = 0 or x = 1

So the points of inflection are (0,0) and (1, -1).

(viii) To sketch the graph of f(x), we can use the information we have gathered so far.

At x = 0, f(x) has a relative minimum of 0 and is concave down. At x = 3/2, f(x) has a relative minimum of -27/16 and is concave up. The point (1, -1) is a point of inflection.

Based on this information, we can sketch a graph of f(x) that looks like this:

```

|

|

|

|

|

|

|

--------o------------o-------

0 3/2 x-axis

```

The graph is a "U" shape that opens upward, with a relative minimum at (3/2, -27/16) and a point of inflection at (1, -1).