if x < y < z and all three are consecutive non-zero integers, then which of the following must be a positive odd integer?

The simplified form of the combination is (3n² - 5n + 3)/3

Now let's return to the problem at hand. We have the expression ⁿ⁺¹C₃ − ⁿ⁻¹C₃, which involves two combinations with slightly different values of n. To simplify this expression, we need to first expand each combination using the formula we just discussed.

Starting with ⁿ⁺¹C₃, we have:

ⁿ⁺¹C₃ = (n + 1)!/(3!(n - (3 + 1))!)

We can simplify this further by noting that (n - 4)! is the same as (n - 3 - 1)! and pulling out the common factors:

(n + 1)!/(3!(n - 4)!) = (n + 1)(n)(n - 1)/(3 x 2 x 1)

Next, let's expand ⁿ⁻¹C₃:

ⁿ⁻¹C₃ = (n - 1)!/(3!(n - (3 - 1))!)

Again, we can simplify by noting that (n - 2)! is the same as (n - 1 - 1)!:

(n - 1)!/(3!(n - 2)!) = (n - 1)(n - 2)(n - 3)/(3 x 2 x 1)

Now we can substitute these expressions back into the original equation:

ⁿ⁺¹C₃ − ⁿ⁻¹C₃ = (n + 1)(n)(n - 1)/(3 x 2 x 1) - (n - 1)(n - 2)(n - 3)/(3 x 2 x 1)

We can simplify this by finding a common denominator and combining like terms:

[(n + 1)(n)(n - 1) - (n - 1)(n - 2)(n - 3)]/(3 x 2 x 1)

Simplifying further, we can expand the terms:

[(n³ - n) - (n³ - 7n² + 11n - 6)]/(6)

Which gives us:

(6n² - 10n + 6)/6

We can simplify this by factoring out a 2 from each term in the numerator and denominator:

2(3n² - 5n + 3)/2(3)

And finally, we can cancel out the 2s to get:

(3n² - 5n + 3)/3

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Complete Question:

Simplify:  

ⁿ⁺¹C₃ − ⁿ⁻¹C₃

Answer:

The sequence is defined as follows:

a1 = 5

an = an-1 - 5, for n > 1

Using this definition, we can find the first four terms of the sequence as follows:

a1 = 5

a2 = a1 - 5 = 5 - 5 = 0

a3 = a2 - 5 = 0 - 5 = -5

a4 = a3 - 5 = -5 - 5 = -10

Therefore, the first four terms of the sequence are: 5, 0, -5, -10.

Answer: 24

Step-by-step explanation:

Surface Area of a rectangular prism = 2 (lh +wh + lw ) Square units.

                                                              = 2[(2*2)+(2*2)+(2*2)]

                                                               = 2[4+4+4]

                                                                =2[12]

                                                                  24

The rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.

The given formula for the amount of bacteria present in a medium after t hours is:

[tex]a(t) = 16e^ (0.32t)[/tex]

Now, we need to find out at what rate the amount of bacteria is changing after 12 hours.

This means we need to find out the derivative of a(t) with respect to t and then substitute t = 12 in the derivative formula to get the rate of change of bacteria after 12 hours.

Differentiating the given formula for a(t) with respect to t, we get:

a'(t) = [tex]16(0.32)e^(0.32t)[/tex]

On substituting, t = 12, we get

a'(12) = [tex]16(0.32)e^ (0.32X12)[/tex]

On solving this, we get a'(12) = 21.698.

Therefore, the rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.

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A unity feedback control system is characterized by an open-loop transfer function G(s)H(s)=k/s(s+10). The system gain k so that the system will have a damping ratio of 0.5 is 4. The corresponding rise time is 2.3s, settling time is 4.4s, and peak overshoot is 0.26.

To determine the system gain k so that the system will have a damping ratio of 0.5, we can use the equation  ζ=1/2ωn. Rearranging the equation yields k=2ωn2. Using the damping ratio of 0.5, this yields k=4.
For this value of k, the rise time, settling time, and peak overshoot can be determined using the following equations:
Rise Time: Tr=4.6/ωn
Settling Time: Ts=4.4/ζωn
Peak Overshoot: Mp=e^(-ζπ/√1-ζ^2)
Given that the damping ratio is 0.5 and the system is subjected to a step input of 1V, we can determine the rise time, settling time, and peak overshoot.
Rise Time: Tr=4.6/ωn=4.6/2=2.3s
Settling Time: Ts=4.4/0.5ωn=4.4/1=4.4s
Peak Overshoot: Mp=e^(-0.5π/√1-0.5^2)=0.26

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In order to determine the rate at which the searchlight is rotating when the man is 15 ft from the point on the path closest to the searchlight, the following steps should be performed in this order

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Answer:

I zont know i need points

Step-by-step explanation:

Yea im srry

a) Graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380

b) The maximum profit of $5200 achieved.

Selling profit is the profit that a business makes on the sale of its products or services. It is the difference between the selling price and the cost of product.

a) Let x be the number of boards of mahogany sold and y be the number of boards of black walnut sold. Then, the constraints of the problem can be represented by the following system of inequalities:

x ≥ 0 (non-negative constraint)

y ≥ 0 (non-negative constraint)

x ≤ 260 (maximum number of mahogany boards available)

y ≤ 320 (maximum number of black walnut boards available)

x + y ≤ 380 (maximum number of boards that can be sold)

To graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380 on a coordinate plane and shade the feasible region that satisfies all of the constraints. The feasible region is the area that is bounded by these lines and includes the origin (0, 0).

b) The profit function P(x, y) can be defined as follows:

P(x, y) = 20x + 6y

To maximize the profit, we need to find the values of x and y that satisfy all of the constraints and maximize the profit function P(x, y).

One way to do this is to use the corner-point method. We can evaluate the profit function at each of the corners of the feasible region and find the corner that gives the maximum profit.

The corners of the feasible region are (0, 0), (0, 320), (260, 0), and (120, 260).

P(0, 0) = 0

P(0, 320) = 6(320) = 1920

P(260, 0) = 20(260) = 5200

P(120, 260) = 20(120) + 6(260) = 4720

Therefore, the maximum profit of $5200 can be achieved by selling 260 boards of mahogany and 0 boards of black walnut.

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Answer:

-----------------------------------

The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0.

To find the y-intercept, substitute 0 for x in the equation y = 4x + 2:

The y-intercept is 2 and matching choice is C.

Answer:

2

Step-by-step explanation:

We need to find out the y intercept of the given equation which is y = 4x + 2 .

y intercept is a point where the graph of the equation cuts the y axis. So at y intercept x coordinate will be 0 , so plug in x = 0 , in the given equation as ;

y = 4x + 2

y = 4(0) + 2

y = 0 + 2

y = 2

Therefore the y intercept of the line is 2 . ( option C )

Answer:

see explanation

Step-by-step explanation:

(a)

the scale factor is the ratio of corresponding sides, shape B to shape A

scale factor = [tex]\frac{12}{15}[/tex] = [tex]\frac{4}{5}[/tex]

(b)

t is then [tex]\frac{4}{5}[/tex] of 7 = [tex]\frac{4}{5}[/tex] × 7 = [tex]\frac{28}{5}[/tex]

The right answer is provided by: Taking into account the hypothesis tested and the p-value of the test.

There is enough proof to conclude that using drugs X and Y results in different mean cholesterol levels.

What theories are being tested?

If there is no difference in the cholesterol levels, then the null hypothesis is tested, which is:

H₀:Hₓ-H

It is determined whether there is a difference at the alternative hypothesis, so:

H₀:Hₓ-H≠0

The p-value of 0.003 0.01 leads us to reject the null hypothesis and determine that there was a difference, hence the following is the right response:

There is enough proof to conclude that using drugs X and Y results in different mean cholesterol levels.

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Answer:

Step-by-step explanation:

Regular price:$19

60 per cent of $19 is 11.4 dollars.

but if ypu are looking for how much did she pay then it is..

19-11.4=$7.6

The rounding of the number to 1 significant figure is-

216 × 876 = 180000

216 × 876

This number can be written in form of rounding to 1 significant figure as;

200 × 900 = 180000

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The family will use 3,600 fewer gallons of water with the new showerheads in a 30-day period.

A family of five recently replaced its 5-gallon-per-minute showerheads with water-saving 2-gallon-per-minute showerheads. Each member of the family averages 8 minutes in the shower per day. In a 30-day period, the family will use 3,600 fewer gallons of water with the new showerheads. How many fewer gallons of water will the family use with the new showerheads?

Calculate the number of gallons used per person per shower before:

5 gal/min x 8 min = 40 gallons per person per shower.

The number of gallons used per person per shower after:

2 gal/min x 8 min = 16 gallons per person per shower.

The number of gallons used per person per day before:

40 gallons x 1 shower = 40 gallons per person per day.

The number of gallons used per person per day after:

16 gallons x 1 shower = 16 gallons per person per day

The number of gallons used per family per day before:

40 gallons x 5 people = 200 gallons per day

The number of gallons used per family per day after:

16 gallons x 5 people = 80 gallons per day

The number of gallons used per month before:

40 gallons x 5 people x 30 days = 6,000 gallons per month

The number of gallons used per month after:

16 gallons x 5 people x 30 days = 2,400 gallons per month

The family will use 6,000 - 2,400 = 3,600 fewer gallons of water with the new showerheads in a 30-day period. Therefore, the answer is 3,600.

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In response to the stated question, we may state that As a result, the formula for the sequence's nth term is: [tex]a_n = a _1 * r^{(n-1) (n-1)}[/tex]

In mathematics, a sequence is an ordered list of items. Elements can be numbers, functions, or other mathematical objects. A series is commonly expressed by putting the phrases in parentheses and separating them with commas. A natural number series, for example, can be denoted as: (1, 2, 3, 4, 5, ...) Similarly, the even number series is labelled as follows: (2, 4, 6, 8, 10, ...) A series can be finite or infinite depending on whether it has a finite or infinite number of words.

a. Since the difference between subsequent terms is constant, the series 5, 7, 8,... is an arithmetic sequence. The usual difference is 2, specifically. As a result, the formula for the sequence's nth term is:

[tex]a_n = a_1 + (n-1)d[/tex]

where a 1 represents the first term, d represents the common difference, and n represents the term number. Using the provided values, we get:

[tex]a_n = 5 + (n-1)2\\a_n = 2n + 3[/tex]

b. Since the ratio between subsequent terms is constant, the series 6, 30, 150,... is a geometric sequence. The common ratio is 5, specifically. As a result, the formula for the sequence's nth term is:

[tex]a_n = a _1 * r^{(n-1) (n-1)}[/tex]

where a 1 denotes the first term, r the common ratio, and n the term number

[tex]a_n = 6 * 5^{(n-1) (n-1)}[/tex]

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Answer:

Cost price (CP) = Rs20

% Profit = 10%

Selling price (SP) = y

By formula:

% Profit = (Profit/CP) x 100%

But Profit= SP - CP

Therefore;

% Profit = [(SP-CP)/C.P)] x 100%

Substituting, we have:

10% = [(y-20)/20] x 100%

10/100 = (y-20)/20

1/10 = (y-20)/20

1 = (y-20)/2

Cross multiply

y - 20 = 2

y = 2 + 20

y = 22

Therefore, the Sale price is Rs 22

A theorem which these transformations reveal include the following: theorem of intersecting secants.

In Mathematics and Geometry, the theorem of intersecting secants states that when two lines intersect outside a given circle, the measure of the angle formed by these intersecting lines is equal to one-half (½) of the difference of the two (2) arcs it intercepts.

By applying the theorem of intersecting secants, the value of any of the angle subtended by the intersecting lines can be calculated by using the following mathematical equation:

m∠a = One-half(y – x).

m∠a = ½(y – x).

Where:

x and y represent the angles formed by the intersecting lines.

Therefore, the theorem of intersecting secants describe these set of transformations.

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Answer:

  r ≈ 44.25 cm

Step-by-step explanation:

You want the hypotenuse of a right triangle in which the side adjacent to the angle of 57° is 24.1 cm.

The cosine relation is ...

  Cos = Adjacent/Hypotenuse

  cos(57°) = (24.1 cm)/r

Solving for r gives ...

  r = (24.1 cm)/cos(57°) ≈ 44.25 cm

The length r is about 44.25 cm.

__

Additional comment

The calculator shown in the attachment has its angle mode set to degrees.

<95141404393>

Answer:

The point z = 3+4i is plotted as a blue dot, and the two square roots are plotted as a red dot and a green dot. The magnitudes of z and its square roots are shown by the radii of the circles centered at the origin.

Step-by-step explanation:

qrt(z) = +/- sqrt(r) * [cos(theta/2) + i sin(theta/2)]

where r = |z| = magnitude of z and theta = arg(z) = argument of z.

Calculate the magnitude of z:

|r| = sqrt((3)^2 + (4)^2) = 5

And the argument of z:

theta = arctan(4/3) = 0.93 radians

Now, find the two square roots of z:

sqrt(z) = +/- sqrt(5) * [cos(0.93/2) + i sin(0.93/2)]

= +/- 1.58 * [cos(0.47) + i sin(0.47)]

= +/- 1.58 * [0.89 + i*0.46]

Using a calculator, simplify this expression to:

sqrt(z) = +/- 1.41 + i1.41 or +/- 0.2 + i2.8

Therefore, the total number of ways to select 2 cooked and 3 uncooked eggs is 210 ways.

In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In other words, a combination is a selection of items from a larger set, without regard to the order in which they are chosen. The number of combinations of size r that can be chosen from a set of n elements is denoted by the symbol C(n, r), and is given by the formula:

C(n, r) = n! / (r! * (n-r)!)

where n! (pronounced "n factorial") is the product of all positive integers up to n, and r! is the product of all positive integers up to r.

Here,

a) To find the number of ways to select 2 cooked or 3 uncooked eggs, we need to use the combination formula, which is:

ⁿCₓ= n! / r!(n-r)!

where n is the total number of eggs, r is the number of eggs we want to select, and ! represents factorial.

For 2 cooked eggs, we can select them in:

⁷C₂ = 7! / 2!(7-2)! = 21 ways

For 3 uncooked eggs, we can select them in:

⁵C₃ = 5! / 3!(5-3)! = 10 ways

Therefore, the total number of ways to select 2 cooked or 3 uncooked eggs is:

21 + 10 = 31 ways

b) To find the number of ways to select 2 cooked and 3 uncooked eggs, we need to use the product rule of counting. That is, we need to multiply the number of ways to select 2 cooked eggs and 3 uncooked eggs.

The number of ways to select 2 cooked eggs is:

⁷C₂ = 21 ways

The number of ways to select 3 uncooked eggs is:⁵C₃ = 10 ways

Therefore, the total number of ways to select 2 cooked and 3 uncooked eggs is:

21 x 10 = 210 ways

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Answer:

7/10

59/100

11/20

Step-by-step explanation:

59/100=59/100

7/10=70/100

11/20=55/100

Answer:

-10 or -4

Step-by-step explanation:

the absolute value of -10 + 7 is 3

the absolute value of -4 + 7 is 3

absolute value just removes the negative sign of what is inside of it.

The new coordinates of the two points after rotating the parallel lines 180 degrees clockwise are (2, -7) and (-8, 5).

What are parallel lines?

In geometry, parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet.

If the set of parallel lines contains the points (-2, 7) and (8, -5), then the two lines are parallel to each other and have the same slope. We can find the slope of the line that passes through these two points using the slope formula:

slope = (y2 - y1) / (x2 - x1)

slope = (-5 - 7) / (8 - (-2))

slope = -12 / 10

slope = -6 / 5

So, the equations of the two parallel lines are:

y - 7 = (-6 / 5)(x + 2) --- equation 1

y + 5 = (-6 / 5)(x - 8) --- equation 2

To rotate the lines 180 degrees clockwise, we need to negate both the x and y coordinates of the points on the lines. That is, we need to replace each point (x, y) with the point (-x, -y).

So, after the rotation, the new coordinates of the two points will be:

(-(-2), -7) = (2, -7) --- for point (-2, 7)

(-(8), -(-5)) = (-8, 5) --- for point (8, -5)

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The value of x for each circle is calculated as follows:

1. x = 148°

3. x = 82°

4. x = 135°

5. x = 7

8. x = 6

9. x = 13

The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.

1. Given, SR = ST

then, arcSR = arcST = x°

and arcRT = 64°

Sum of arc,

x + x +64 =360

x = 148°

3. Given, JK=LM

then, arcJK=arcLM

So, x=82°

4. Given, BA=AC =9

Then, arcBA = arcAC = x°

and arcBC = 90°

Sum of arc,

x + x +90 =360

x = 135°

5. Given, arc are equal then side also equal

2x+4=18

x=7

8. Given that, Circle M similar to circle P, and arc are equal so, side also equal

2x+24=6x

x=6

9. Given that, Circle V similar to circle W,

arcYZ + 198 =360

arcYZ = 162°

Now, arcTU=arcYZ

Then, 9x-78 = 3x

So, x=13

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Answer:

487

or

4870

Step-by-step explanation:

(I might not have done it right)

is it 24,354 / 5 or 2,435 / 5?

Because I got 487

But to check an answer you must multiply

When I multiplied 5 x 487 the answer is 2,435

But 5 x 4870 is 24,350

It has to be one of these

The interest charge per month would be around $[tex]532.02[/tex], and the principal payment per month would be about $[tex]399.09[/tex].

Your hobbies are the things you find enjoyable to do and the things that you prefer to learn more about. A lack of intriguing conversation: Interesting things hold your attention and arouse your curiosity.

The cost of borrowing money is called interest, and it is typically stated as a percent, such as an annualized rate (APR). By utilizing the money, debtors may accrue interest to lenders, or lenders may pay the interest to borrowers.

We will use annuity formula, which is: [tex]P=C(\frac{1-(1+r)^{-n} }{r} )[/tex]

[tex]P[/tex] is [tex]105,000[/tex]

[tex]C[/tex] is what we need

[tex]r[/tex] is the [tex]0.045/12 = 0.00375[/tex]

[tex]n[/tex] is [tex]12*30 = 360[/tex]

[tex]P=C[\frac{1-(1+r)^{-n} }{r} ][/tex]

[tex]105,000=C[\frac{1-(1+0.00375)^{-360} }{0.00375} ][/tex]

[tex]1.5,000=C[197.3612][/tex]

[tex]C=532.02[/tex]

So monthly payment would be around [tex]532.02[/tex]

Now,

With each point purchase, the interest rate goes down by [tex]0.25[/tex]%, so for [tex]2[/tex] points it will be [tex]4.5[/tex]%[tex]- 2(0.25) = 4[/tex]%

Also, since [tex]20[/tex]% down payment, the loan amount would be [tex](0.8)(105,000) = 84,000[/tex].

Putting these values into the annuity formula we have:

[tex]84000=C[\frac{1-(1+.0..33)^{-360} }{0.0033} ][/tex]

[tex]84000=C(210.47)[/tex]

[tex]C=399.09[/tex]

The monthly payment would be around $[tex]399.09[/tex]

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Rs 5.25 can be expressed without decimals as Rs 525/100 or as 5 1/4 (or 5 ¼) using mixed numbers. Both methods allow us to represent the value of Rs 5.25 accurately without using decimal notation.

To express Rs 5.25 without using decimals, we need to convert it into a fraction with a numerator and a denominator. One way to do this is by multiplying both the numerator and the denominator by 100 to eliminate the decimal point.

So, Rs 5.25 can be written as Rs 525/100. This is also known as a fraction in its simplest form, as both the numerator and denominator have no common factors other than 1.

Another way to express Rs 5.25 without decimals is to use mixed numbers. A mixed number is a whole number combined with a fraction. To convert a decimal to a mixed number, we need to find the whole number part and the fraction part.

To find the whole number part, we can simply drop the decimal point and write the digits to the left of it. In this case, the whole number part is 5.

To find the fraction part, we need to subtract the whole number part from the original number and simplify the fraction. In this case, Rs 5.25 can be written as 5 and 1/4 (or 5 ¼), where the fraction part is 1/4.

In conclusion, Rs 5.25 can be expressed without decimals as Rs 525/100 or as 5 1/4 (or 5 ¼) using mixed numbers. Both methods allow us to represent the value of Rs 5.25 accurately without using decimal notation.

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Use the line segment that connects (3, 42.5) and (5, 44) to estimate the vine length after 4 days.

Given the table of values

To do this, we make use of linear interpolation

Such that the points closest to day 4 are (3, 42.5) and (5, 44)

The slope of the line passing through the two points can be found using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) = (3, 42.5) and (x2, y2) = (5, 44).

Thus, we have:

m = (44 - 42.5) / (5 - 3) = 0.75

So, we have

y = 0.75x + b

Using the point (3, 42.5), we get:

42.5 = 0.75(3) + b

Solving for b, we get:

b = 42.5 - 0.75(3)

b = 40.25

So, the equation is

y = 0.75x + 40.25

To estimate the vine length after 4 days, we substitute x = 4 into the equation and solve for y:

y = 0.75 * 4 + 40.25

y = 43.25

Therefore, the vine length after 4 days is estimated to be about 43.25 units.

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Step-by-step explanation:

(a) To find the lower bound for the length of the rectangle, we need to subtract half of the smallest unit of measurement from the given measurement.

The smallest unit of measurement given is 1 mm, so half of that is 0.5 mm.

Therefore, the lower bound for the length of the rectangle is:

64 mm - 0.5 mm = 63.5 mm

(b) The perimeter of a rectangle is calculated by adding up the length of all four sides.

The lower bounds for the length and width of the rectangle are 63.5 mm and 36.5 mm, respectively.

So, the lower bound for the perimeter is:

2(63.5 mm + 36.5 mm) = 2(100 mm) = 200 mm

Answer and explanation below. Credits to Chris, Mechanical engineering college, Master's degree

The function that satisfies the hypotheses of the Extreme Value Theorem on the given interval is given by

B. g(x)=6x^2+3 on 0≤x≤4

D. f(x) = (e^x)/x for 2 ≤ x ≤ 16.

Step 1: State the Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval.

Step 2: Check for continuity and closed interval for each function

A. f(x) = 1/x on −10 ≤ x ≤ 10

The function f(x) = 1/x is continuous on the interval (-10, 0) and (0, 10).

However, since the interval given is [−10, 10], we see that the function is not continuous over the closed interval.

Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.

B. g(x) = 6x^2+3 on 0 ≤ x ≤ 4

The function g(x) is continuous on the interval [0, 4].

Therefore, this function satisfies the hypotheses of the Extreme Value Theorem on the given interval.

C. k(x) = {3x^2+9 for 0 ≤ x < 2, 12x for 2 ≤ x ≤ 10} on 0 ≤ x ≤ 10

The function k(x) is continuous on the interval [0, 2) and (2, 10]. H

However, since the interval given is [0, 10], we see that the function is not continuous over the closed interval.

Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.

D. h(x) = (e^x)/x for 2 ≤ x ≤ 16The function h(x) is continuous on the interval [2, 16].

Therefore, this function satisfies the hypotheses of the Extreme Value Theorem on the given interval.

E. m(x) = 6x^3+x+1 on −4 < x < 3

The function m(x) is continuous on the interval (-4, 3).

However, since the interval given is [-4, ∞), we see that the function is not continuous over the closed interval.

Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.

Therefore, the only function that satisfies the hypotheses of the Extreme Value Theorem on the given interval is given by

B. g(x)=6x^2+3 on 0≤x≤4

D. f(x) = (e^x)/x for 2 ≤ x ≤ 16

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