Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = x^3 − x^2 − 12x + 3, [0, 4]

Answers

Answer 1

well, is the function continuous? yes, is a cubic one, you can graph it if you wish, is continuous all the way, and of course at [ 0, 4] too.

is it differentiable? you can always look at the graph between 0,4 and is a smooth transition line, thus yes, it is differentiable, but let's check anyway,

[tex]\bf \cfrac{dy}{dx}=3x^2-2x-12[/tex] its derivative has no asymptotes and therefore no "cusps", so yes, is differentiable all around.

is f(0) = f(4), let's check

f(0) = 0+0+0+3, f(0) = 3
f(4) = 64 - 16 - 48 + 3, f(4) = 3

yeap

there must then be a "c" value(s) with a horizontal tangent slope, let's check, is really just the critical points.

[tex]\bf \cfrac{dy}{dx}=3x^2-2x-12\implies 0=3x^2-2x-12 \\\\\\ \textit{using the quadratic formula} \\\\\\ x=\cfrac{-(-2)\pm\sqrt{(-2)^2-4(3)(-12)}}{2(3)}\implies x=\cfrac{2\pm\sqrt{4+144}}{6} \\\\\\ x=\cfrac{2\pm 2\sqrt{37}}{6}\implies x=\cfrac{2\pm\sqrt{37}}{3}\impliedby \textit{c's}[/tex]

Related Questions

Suppose 8 out of every 20 students are absent from school less than 5 days a year.Predict how many students would be absent less than 5 days a year out of 40,000 students.

Answers

Answer:
8/20 * 40,000 = 16,000

What percent of 9.2 is 43.7

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if we take 9.2 to be the 100%, what is 43.7 off of it in percentage then?

[tex]\bf \begin{array}{ccll} amount&\%\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 9.2&100\\ 43.7&p \end{array}\implies \cfrac{9.2}{43.7}=\cfrac{100}{p}\implies p=\cfrac{43.7\cdot 100}{9.2}[/tex]

The requried 475% of 9.2 is 43.7, as of the given percentage situation.

What is the percentage?

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
In the question, we are asked to determine the 43.7 is what percent of 9.2.

So, let the percent be x,
x % of 9.2 = 43.7
x/100 × 9.2 = 43.7
x = 43.7/9.2 × 100%
x = 475%

Thus, the requried 475% of 9.2 is 43.7, as of the given percentage situation.

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Find the length and width of a rectangle that has the given perimeter and a maximum area. perimeter: 128 meters

Answers

P = 128 meters = 2W + 2L. We want to maximize the area: A = L*W

Solve 128 meters = 2W + 2L for either W or L:

64 meters = W + L, so W = 64-L
and subst. your result into

A = L*W: A = L*(64-L). Then A(L) = 64L - L^2. You could graph this and find the approx value of L at which A(L) is at its max.

Or, if you know calculus, differentiate A(L) and set the result = to 0. Solve for L.

L + W = 64, so you can subst. your value for L into this eqn to find W.

Triangle ABC has two known angles. Angle A measures 55 degrees. Angle b measures 30 degrees.What is the measure of angle C?

Answers

95 Degrees, 180-55-30=95

The correct measure of angle C is 95 degrees. Hope this helps.

Determine all possible angles θ for equilibrium, 0∘<θ<90∘.

Answers

If the tension of a cable AB and BC is represented as Fab and Fac, using the equilibrium equations equation 1:Fab cos 40 degrees - Fac cos x = 0where x is the angle we are looking for,0.766Fab - Fac cox x = 0equation 2:-200 + Fab sin 40 degress + Fac sin x = 00.643Fab + Fac sin x = 200solving the tension we getFab = 311.04lbFac = 238.26 lb
substitute 30 degrees to x on equation 1 and 2, we getFab = 184.31lb, Fac = 163lb
substitute 45 degrees to x on equations 1 and 2, we getFab = 141.9lb, Fac = 153.77lb
substitute 60 degrees to equation 1 and 2,we getFab = 101.5lb, Fac = 155.55lb
substitute 90 degrees to x on eqn 1 and 2 we getFab = 0lb, Fac = 200lb
There 90 is not a possible answer.
Possible angles are 30 degrees, 45 degrees and 60 degrees

Two small fires are spotted by a ranger from a fire tower 60 feet above ground. The angles of depressions re 11.6° and 9.4°. How far apart are the fires? (The fires are in the same general direction from the tower.)

Answers

The two fires are about 70 feet from each other. The assumption is that the ground is relatively level and that a right triangle will be made with the three points of the triangle being the ranger, the spot on the ground directly beneath the ranger, and the fire itself. So the distance to the first fire will be: Calculate the angle. That will be 90° - 11.6° = 78.4° The distance will be tan(78.4) = X/60 60 tan(78.4) = X 60 * 4.871620136 = X 292.2972082 = X And that's how far the 1st fire is from the ranger's station. Now for the 2nd angle = 90° - 9.4° = 80.6° 60 tan(80.6) = X 60 * 6.040510327 = X 362.4306196 = X And the distance between the two fires will be the difference in distance from the tower, so 362.4306196 - 292.2972082 = 70.13341146 Rounding to 2 significant figures gives 70 feet.

five friends went to a local restaurant and all had the buffet at $6.75 each. the following is the restaurant bill. does it seem reasonable 5($6.75) =&67.50

Answers

No I think not really to sure

Yes because your food had cost $6.75 each for 5 people, so if you had a bill of $67.50 and your concern about is it reasonable just budget it out.

What is the meaning of the term theorem

Answers

There are 4 terms in the world of *Mathematical proof*
Lemma, Proposition, Corollary and Theorem.
There is no difference between a lemma,
proposition, theorem, or corollary - they are all claims waiting to be proved. However, we use these terms to suggest different levels of importance and difficulty. A lemma is an easily proved claim which is helpful for proving other propositions and theorems, but is usually not particularly interesting in
its own right. A proposition is a statement which is interesting in its own right, while a theorem is a more important statement than a proposition which says something definitive on the subject, and often takes more effort to prove than a proposition or lemma. A corollary is a quick consequence of a proposition or theorem that was proven recently

Please need help as quick as you can . What is the slope of the line that passes through the points (16, –1) and (–4, 10)? A. -20/11 B. -11/20 C. 11/20 D. 20/11

Answers

[tex]\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 16}}\quad ,&{{ -1}})\quad % (c,d) &({{ -4}}\quad ,&{{ 10}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{10-(-1)}{-4-16}\implies \cfrac{10+1}{-4-16} \\\\\\ -\cfrac{11}{20}[/tex]

PLEASE HELP 15 POINTS WILL GIVE BRAINLIEST The following formula, F = ma, relates three quantities: Force (F), mass (m), and acceleration (a). A: Solve this equation, F = ma for a. B If F = -24 units and m = 10 units, what is the acceleration, a? Use the equation from Part (a) to answer the question. C: If F = 24 units and a = 12 units, what is the mass, m? Use the equation, F = ma, to plug in the known values and solve for m

Answers

Part A

The expression "ma" is the same as "m*a" (m times a). To isolate 'a', we divide both sides by m. Division is the inverse operation of multiplication. Think of it as undoing multiplication

F = ma
F = m*a
F/m = m*a/m
F/m = a
a = F/m

Answer: a = F/m
Note: the slash "/" without quotes means "divide"

===========================================
Part B

Use the result from part A. We will plug F = -24 and m = 10 into that equation

a = F/m
a = -24/10
a = -2.4

Answer: -2.4

===========================================
Part C

We will use the equation F = m*a

F = m*a
24 = m*12 ... plug in F = 24 and a = 12
24/12 = m*12/12
2 = m
m = 2

Answer: 2

Find the area of the specified region shared by the circles r = 4 and r = 8sin

Answers

ind the area shared by the circles r=2cos(theta) and r=2sin(theta).

i know the general formula for finding the area, but i don't know which is the outside one, and which is the inside circle. but i've tried both ways and am still not getting the right answer.

so let's just say i'll try it with integral .5(4cos(x)^2-.5(4sin(x)^2)
i can factor out a 2, giving me cos(x)^2-sin(x)^2. the integral of that, i believe, can be expressed as 4sin(2x). now i figured the limits of integration were from 0 to pi/4, because those are the two places the circles intersect. so evaluating there, i would get 4-0=4. but I've been told the answer is pi/2 -1. so where am i going wrong, because i'm way off.

The GCF of any two even numbers is always even. true or false?

Answers

your answer is false.

True because if you use prime factorization of even numbers you will always get a 2. If a number has 2 as a factor then it is even.

the weight of an object on the moon is about .167 of its weight on Earth. How much does a 180 lb astronaut weigh on the moon?

Answers

[tex]\bf \begin{array}{ccll} \stackrel{lbs}{earth}&\stackrel{lbs}{moon}\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 1&0.167\\ 180&x \end{array}\implies \cfrac{1}{180}=\cfrac{0.167}{x}\implies x=\cfrac{180\cdot 0.167}{1}[/tex]

The astronaut will weigh less on the moon. Determine how much the scale will indicate by multi. 180 lb by 0.167: (180 lb)(0.167) = 30.06 lb.

The driver of a car traveling at 54ft/sec suddenly applies the brakes. The position of the car is s=54t-3t^2, t seconds afyer the driver applies the brakes.
How many seconds after the driver applies the brakes does the car come to a stop

Answers

s = 60t - 3t^2
v = ds/dt = 60 - 6t

when it comes to rest, v = 0,
so t = 10 sec

now distance travelled = time taken x average velocity
= 10 x (60 + 0)/2
= 300 ft

After time [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.

What is time?

" Time is defined as the measurable slot of period in which required action is done."

Formula used

[tex]y = x^{n} \\\\\implies \frac{dy}{dx} = nx^{n-1}[/tex]

According to the question,

Position of the car [tex]'s' = 54t - 3t^{2}[/tex]

[tex]'s'[/tex] represents the distance

[tex]'t'[/tex] is the time in seconds

When driver applies break [tex]v= 0[/tex],

[tex]v = \frac{ds}{dt}[/tex]

Calculate the first derivative with respect to time we get,

[tex]'s' = 54t - 3t^{2}\\\\\implies \frac{ds}{dt} = 54- 6t[/tex]

As [tex]\frac{ds}{dt} =0[/tex] we get the required time as per given condition,

[tex]54-6t =0\\\\\implies 6t =54\\\\\implies t = 9[/tex]

Hence, after time [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.

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Toby exercises 14 hours a week. John exercises 20% more than Toby and Jenny exercises two more hours than John. Which expression represents how much Jenny exercises? (w represents weeks)

Answers

John exercised 20% more than Toby. So the number of hours that John exercises per week is 14 + 0.2(14) = 14 + 2.8 = 16.8.

Jenny exercises two more hours than John. So the number of hours that Jenny exercises per week is 16.8 + 2 = 18.8.

So an expression representing the number of hours that Jenny exercises after w weeks is 18.8w.

the answer is 18.8w

Toby 14w

John 14(1.2)w

Jenny 14(1.2)w + 2w

thus, Jenny exercise

14(1.2)w + 2w

16.8w + 2w

18.8w

For the x-values 1, 2, 3, and so on, the y-values of a function form a geometric sequence that increases in value. What type of function is it? A. Exponential growth B. Decreasing linear C. Increasing linear D. Exponential decay

Answers

Answer:

Option A: Exponential growth

Step-by-step explanation:

If for values of x values of y increases and forms a geometric sequence then it would be an exponential growth function because geometric sequence is an exponential function and since, it is increasing hence, an exponential growth.

Option B is incorrect because because y is not decreasing

Option C and D are incorrect because geometric sequence can never be linear since, it gives common ratio.

Final answer:

The function described, where y-values form an increasing geometric sequence as x-values increase, is characterized as Exponential growth. This is consistent with an exponential curve where the rate of growth is proportional to the current value, which aligns with the behavior of a geometric sequence.

Explanation:

For the given scenario where the y-values of a function form a geometric sequence that increases for the x-values 1, 2, 3, and so on, the type of function is described as Exponential growth. This is because in a geometric sequence each term after the first is found by multiplying the previous term by a constant called the common ratio, which is analogous to the exponential function where the growth rate of the value is proportional to its current value. The more the x-value increases, the higher and faster the y-value grows, following the pattern of an exponential growth curve.

Answer A. Exponential growth fits the description as it involves an increasing geometric sequence. Answer B. Decreasing linear refers to a linear function that decreases with each increment in x, which does not describe the function in question. Similarly, Answer C. Increasing linear describes a function that increases at a constant rate, not geometrically. Answer D. Exponential decay would imply the y-values decrease as x increases, which is also not the case described.

a construction crew Must build 4 miles of road in one week on Monday they build 1/2 mile of road in one week on Tuesday they build 1/3 mile of road how many more miles of road are left to build

Answers

To solve this, you need to make the amount they built on Monday and the amount they built on Tuesday compatible with each other so you can add them together to subtract that amount from 4.

Start by multiplying 1/2 by 3, and 1/3 by 2.

This gives you 3/6 and 2/6. Now add them together:

3/6 + 2/6 = 5/6

Finally, subtract this amount from 4:

4 - 5/6 = 3 1/6 (19/6 works too)

Hope this helps! :)

4 - 1/2 - 1/3
24/6 - 3/6 - 2/6 = 19/6 = 3 1/6

If s = {r, u, d} is a set of linearly dependent vectors. if x = 5r + u + d, determine whether t = {r, u, x} is a linearly dependent set

Answers

Consider any arbitrary linear combination of the vectors [tex]\mathbf r,\mathbf u\,\mathbf x[/tex]. We have

[tex]c_1\mathbf r+c_2\mathbf u+c_3\mathbf x=c_1\mathbf r+c_2\mathbf u+c_3(5\mathbf r+\mathbf u+\mathbf d)[/tex]
[tex]=(c_1+5c_3)\mathbf r+(c_2+c_3)\mathbf u+c_3\mathbf d[/tex]
[tex]=c_4\mathbf r+c_5\mathbf u+c_6\mathbf d[/tex]

We know [tex]\mathbf r,\mathbf u,\mathbf d[/tex] are linearly dependent, which means there must exist some choice of not all zero constants [tex]c_4,c_5,c_6[/tex] such that the combination above gives the zero vector. So [tex]T=\{\mathbf r,\mathbf u,\mathbf x\}[/tex] is a set of linearly dependent vectors.

How much do parents pay per day? If a childcare center charges $150 per week during the school year. In the summer, parents can pay a prorated amount based on the number of days of attendance.

Answers

During the school year, children usually go to the childcare 5 days per week.
Knowing that parents pay $150 per week, we can divide this amount by the number of days children go to the childcare in order to find the daily payment.

This is done as follows:
daily payment = $150 / 5 = 30$ per day

Write the equation of a line, in general form , that has an slope of zero and a y-intercept of (0, -6).

Answers

Use the slope-intercept form.
y = mx + b,

where m = slope and b = y-intercept.
Your y-intercept is -6, so b = -6.
The slope is zero, so m = 0

y = mx + b

y = 0x + (-6)

y = -6

Mr. Carandang sold a total of 1,790 prints of one of his drawings. Out of all 1,273 unframed prints that he sold, 152 were small and 544 were medium-sized. Out of all of the framed prints that he sold, 23 were small and 42 were extra large. Of the large prints that he sold, 188 were framed and 496 were unframed. Small Medium Large Extra Large Total Framed 23 264 188 42 ? Unframed 152 544 496 81 1,273 Total 175 808 684 123 1,790 Which number is missing from the two-way table?'

Answers

framed unframed total
small 23 152 175
medium 264 544 808
large 188 496 684
ex-large 42 81 123
total 517 1273 1790

The missing number in the two-way table is the total number of framed prints, which is 517. This is calculated by subtracting the total unframed prints from the total prints sold and then summing the known framed prints.

To find the missing number of prints in the table, we need to determine the total number of framed prints sold by Mr. Carandang. We know the following from the table:

Total prints sold: 1,790Unframed prints: 1,273

To find the total number of framed prints:

Total framed prints = Total prints - Total unframed prints

Total framed prints = 1,790 - 1,273 = 517

Now we need to sum the known framed prints:

Framed small: 23Framed medium: 264Framed large: 188Framed extra large: 42

The total of known framed prints is:

23 + 264 + 188 + 42 = 517

Thus, the total framed prints value was missing and it is 517.

Eric has 4 bags of 10 marbles and 6 single marbles . how many marbles does eric have

Answers

He has 46 marbles in total

First, you would multiply the number of bags by the quantity in each.
4 x 10 = 40

Next, you would add the single marbles.
40 + 6 = 46

Your answer would be 46.

I hope this helps!

I keep getting 13. It's supposed to be 3. Can you show me how it's done? 3(a-5) = -6

Answers

First, divide both sides by 3. I think your mistake was that you were multiplying -3. Please note that in order to cancel something out you need to do the reverse operation. The reverse operation of subtraction is addition and the reverse of multiplication is division.

3(a - 5) = -6
a - 5 = -2

Now add 5 to both sides

a - 5 = -2
+5 +5
a = 3

Hope this helps!

3(a-5) = -6

Start by distributing...

3a - 15 = - 6

Add 15 to both sides...

3a = 9

Divide both sides by 3 to get a by its self

a = 3

In Miss Marshalls classroom 6/7 of the students play sports of the students who play sports for fifth also play instruments if there are 35 students in her class how many play sports and instruments

Answers

6/7 time 4/5 = 24/35
24 students do both.

Find the rate of change of the area of a square with respect to the length z , the diagonal of the square. what is the rate when z=2?

Answers

The diagonal of a square is equal to the side x times square root of 2, xSqrt(2)
z = xSqrt(2), its rate of change is just Sqrt(2)

The rate of change of Area, A with respect to the diagonal length, z and the rate of change when z = 2 is :

[tex]\frac{dA}{dz} = z [/tex]

[tex]\frac{dA}{dz} = 2 \: when \: z = 2 [/tex]

The area of a square is rated to its diagonal thus :

Area of square = A

Length of diagonal = z

The relationship between Area and diagonal of a square is : [tex]A \: = 0.5 {z}^{2} [/tex]

The rate of change of area with respect to the length, z of the square's diagonal ;

This the first differential of Area with respect to z

[tex] \frac{dA}{dz} = 2(0.5)z \: = z[/tex]

Therefore, the rate of change of area, A with respect to the length, z of the diagonal is [tex]\frac{dA}{dz} = z [/tex]

The rate of change [tex]\frac{dA}{dz} [/tex] when z = 2 can be calculated thus :

Substitute z = 2 in the relation [tex]\frac{dA}{dz} = z [/tex]

Therefore, [tex]\frac{dA}{dz} = 2 [/tex]

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Find the difference: 16.25 - 7.92

Answers

Your answer is going to be 8.33

the difference is 8.33.
give me the best plz

what is the least common denominator for 5/6 and 3/8

Answers

The least common denominator is 24.

F(x, y) = x2 + y2 + 4x − 4y, x2 + y2 ≤ 81 find the extreme values of f on the region described by the inequality.

Answers

Take partial derivatives and set them equal to 0:

[tex]\nabla F(x,y)=\langle2x+4,2y-4\rangle=\mathbf 0[/tex]

We find one critical point within the boundary of the disk at [tex](x,y)=(-2,2)[/tex]. The Hessian matrix for this function is

[tex]\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2&0\\0&2\end{bmatrix}[/tex]

which is positive definite, and incidentally independent of [tex]x[/tex] and [tex]y[/tex], so [tex]F(x,y)[/tex] attains a minimum [tex]F(-2,2)=-8[/tex].

Meanwhile, we can parameterize the boundary by [tex]\mathbf r(t)=\langle9\cos t,9\sin t\rangle[/tex] with [tex]0\le t\le2\pi[/tex], which gives

[tex]F(x,y)=F(x(t),y(t))=f(t)=81\cos^2t+81\sin^2t+36\cos t-36\sin t=81+36(\cos t-\sin t)[/tex]

with critical points at

[tex]f'(t)=0\implies 36(-\sin t-\cos t)=0\implies\sin t+\cos t=0\implies t=\dfrac{3\pi}4,\dfrac{7\pi}4[/tex]

At these points, we get

[tex]f\left(\dfrac{3\pi}4\right)=81-36\sqrt2\approx30.0883[/tex]
[tex]f\left(\dfrac{7\pi}4\right)=81+36\sqrt2\approx131.9117[/tex]

so we attain a maximum only when [tex]t=\dfrac{7\pi}4[/tex], which translates to [tex](x,y)=\left(\dfrac9{\sqrt2},-\dfrac9{\sqrt2}\right)[/tex].

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are: -8 and 131.91

The given parameters are:

[tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Find the gradient of F(x,y)

[tex]\mathbf{f_x(x) = 2x + 4}[/tex]

[tex]\mathbf{f_y(y) = 2y - 4}[/tex]

Set to 0, to solve for x and y

[tex]\mathbf{2x + 4 = 0}[/tex]

[tex]\mathbf{2x = -4}[/tex]

[tex]\mathbf{x = -2}[/tex]

[tex]\mathbf{2y - 4 = 0}[/tex]

[tex]\mathbf{2y = 4}[/tex]

[tex]\mathbf{y = 2}[/tex]

So, the critical point is:

[tex]\mathbf{(x,y) = (-2,2)}[/tex]

Also, we have: [tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Calculate the gradients

[tex]\mathbf{g_x = 2x}[/tex]

[tex]\mathbf{g_y = 2y}[/tex]

Equate the gradients as follows:

[tex]\mathbf{f_x = \lambda \cdot g_x}[/tex]

[tex]\mathbf{f_y = \lambda \cdot g_y}[/tex]

So, we have:

[tex]\mathbf{2x + 4 = \lambda \cdot 2x}[/tex]

[tex]\mathbf{2y - 4 = \lambda \cdot 2y}[/tex]

Make [tex]\mathbf{\lambda}[/tex] the subject, in the above equations

[tex]\mathbf{\lambda = \frac{x + 2}{x}}[/tex]

[tex]\mathbf{\lambda = \frac{y - 2}{y}}[/tex]

Equate the above equations, so we have:

[tex]\mathbf{\frac{x + 2}{x} = \frac{y - 2}{y} }[/tex]

Cross multiply

[tex]\mathbf{xy + 2y = xy - 2x}[/tex]

Subtract xy from both sides

[tex]\mathbf{2y =- 2x}[/tex]

Divide both sides by 2

[tex]\mathbf{y =- x}[/tex]

Substitute [tex]\mathbf{y =- x}[/tex] in [tex]\mathbf{x^2 + y^2 = 81}[/tex]

[tex]\mathbf{x^2 + (-x)^2 = 81}[/tex]

[tex]\mathbf{x^2 + x^2 = 81}[/tex]

[tex]\mathbf{2x^2 = 81}[/tex]

Divide both sides by 2

[tex]\mathbf{x^2 = \frac{81}{2}}[/tex]

Take square roots

[tex]\mathbf{x = \±\frac{9}{\sqrt2}}[/tex]

Rationalize

[tex]\mathbf{x = \±\frac{9\sqrt2}{2}}[/tex]

Recall that: [tex]\mathbf{y =- x}[/tex]

So, we have:

[tex]\mathbf{y = \±\frac{9\sqrt2}{2}}[/tex]

The ordered pairs are:

[tex]\mathbf{(x,y) = \{(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Hence, the points are:

[tex]\mathbf{(x,y) = \{(-2,2),(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Substitute these values in [tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{F(2,2) = (-2)^2 + 2^2 + 4(-2) - 4(2) =-8}[/tex]

[tex]\mathbf{F(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}) = (\frac{9\sqrt2}{2})^2 + (-\frac{9\sqrt2}{2})^2 + 4(\frac{9\sqrt2}{2}) - 4(-\frac{9\sqrt2}{2}) =131.91}[/tex]

[tex]\mathbf{F(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2}) = (-\frac{9\sqrt2}{2})^2 + (\frac{9\sqrt2}{2})^2 + 4(-\frac{9\sqrt2}{2}) - 4(\frac{9\sqrt2}{2}) =30.09}[/tex]

Hence, the extreme values of the function are: -8 and 131.91

Answers

first do f(a+h)-f(a)
[tex] \frac{5a+5h}{a+h-1} - \frac{5a}{a-1} [/tex]

make the denominators the same and simplify:
[tex] \frac{(5a+5h)(a-1)}{(a+h-1)(a-1)} - \frac{5a(a+h-1)}{(a+h-1)(a-1)} [/tex]

expand: [tex] \frac{5a^2-5a+5ah-5h-(5a^2+5ah-5a)}{(a+h-1)(a-1)} [/tex]
simplify: [tex] \frac{-5h}{(a+h-1)(a-1)} [/tex]

now divide it by h, you have: [tex] \frac{-5}{(a+h-1)(a-1)} [/tex]

[tex]\bf \cfrac{f(a+h)-f(a)}{h}\implies \cfrac{\frac{5a+5h}{a+h-1}~~-~~\frac{5a}{a-1}}{h}\impliedby \stackrel{LCD}{(a+h-1)(a-1)} \\\\\\ \cfrac{\frac{(a-1)(5a+5h)~~-~~(a+h-1)(5a)}{(a+h-1)(a-1)}}{h} \\\\\\ \cfrac{\frac{5a^2+5ah-5a-5h~~-~~(5a^2+5ah-5a)}{(a+h-1)(a-1)}}{h}[/tex]

[tex]\bf \cfrac{\frac{\underline{5a^2+5ah-5a}-5h~~\underline{-5a^2-5ah+5a}}{(a+h-1)(a-1)}}{h}\implies \cfrac{\frac{-5h}{(a+h-1)(a-1)}}{h} \\\\\\ \cfrac{\frac{-5h}{(a+h-1)(a-1)}}{\frac{h}{1}}\implies \cfrac{-5h}{(a+h-1)(a-1)}\cdot \cfrac{1}{h} \\\\\\ \cfrac{-5\underline{h}}{\underline{h}(a+h-1)(a-1)}\implies \cfrac{-5}{(a+h-1)(a-1)}[/tex]

A car manufacturer claims that, when driven at a speed of 50 miles per hour on a highway, the mileage of a certain model follows a normal distribution with mean μ = 30 miles per gallon and standard deviation σ= 4 miles per gallon.

Answers

84 miles per gallon maybe

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