Darboux's Theorem: Let f be a real valued function on the closed interval [a,b]. Suppose f is differentiable on [a,b]. Then f′ satisfies the intermediate value property.
What is the intermediate value property?
Give an example of a function defined on [a,b] that is not the derivative of any function on [a,b]
Give an example of a differentiable function f on [a,b] such that f′ is not continuous.
Present a proof of Darboux's theorem.
Answers
The answer to the question :
Darboux's Theorem: Let f be a real-valued function on the closed interval [a,b]. Suppose f is differentiable on [a,b]. Then f′ satisfies the intermediate value property.
What is the intermediate value property?
Give an example of a function defined on [a,b] that is not the derivative of any function on [a,b]
Give an example of a differentiable function f on [a,b] such that f′ is not continuous.
Present proof of Darboux's theorem. is given below:
Explanation:
The intermediate value property refers to the property that a continuous function takes all values between its maximum and minimum value in a closed interval. The intermediate value property states that if f is continuous on the closed interval [a,b], and L is any number between f(a) and f(b), then there exists a point c in (a, b) such that f(c) = L.
For an example of a function defined on [a,b] that is not derivative of any function on [a,b], consider f(x) = |x| on the interval [-1, 1]. This function is not differentiable at x = 0 since the left and right-hand derivatives do not match.
An example of a differentiable function f on [a,b] such that f′ is not continuous is f(x) = x^2 sin(1/x) for x not equal to 0 and f(0) = 0. The derivative f′(x) = 2x sin(1/x) − cos(1/x) for x not equal to 0 and f′(0) = 0. The function f′ is not continuous at x = 0 since f′ oscillates wildly as x approaches 0.
Darboux's Theorem: Let f be a real-valued function on the closed interval [a, b]. Suppose f is differentiable on [a,b]. Then f′ satisfies the intermediate value property.
Proof: Suppose, for the sake of contradiction, that f′ does not satisfy the intermediate value property. Then there exist numbers a < c < b such that f′(c) is strictly between f′(a) and f′(b). Without loss of generality, assume f′(c) is strictly between f′(a) and f′(b).
By the mean value theorem, there exists a number d in (a, c) such that
f′(d) = (f(c) − f(a))/(c − a).
Similarly, there exists a number e in (c, b) such that
f′(e) = (f(b) − f(c))/(b − c).
Now,
(f(c) − f(a))/(c − a) < f′(c) < (f(b) − f(c))/(b − c).
Rearranging terms, we have
(f(c) − f(a))/(c − a) − f′(c) < 0 and (f(b) − f(c))/(b − c) − f′(c) > 0.
Define a new function g on the interval [a, b] by
g(x) = (f(x) − f(a))/(x − a) for x ≠ a and g(a) = f′(a). Then g is continuous on [a, b] and differentiable on (a, b).
By the mean value theorem, there exists a number c in (a, b) such that
g′(c) = (g(b) − g(a))/(b − a) = (f(b) − f(a))/(b − a).
However,
g′(c) = f′′(c), so f′′(c) = (f(b) − f(a))/(b − a).
Since f′′(c) is strictly between (f(c) − f(a))/(c − a) and (f(b) − f(c))/(b − c), we have a contradiction. Therefore, f′ must satisfy the intermediate value property.
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Related Questions
entertain me in some way ill give brainliest
Answers
Answer:
i stupida
Step-by-step explanation:
UvU
the reciprocal of -2 3/7
Answers
A bookstore charges a standard rate for paperback or hardback books. The cost of a paperback book is 4 dollars less than the cost of a hardback book. Recently, one day of sales totaled $1065.10. That day the bookstore sold 51 paperback books and 47 hardback books. Write a system representing this situation. Use the graph to estimate the solution of the system. What does the solution mean in this situation? How much does each type of book cost? Use substitution to verify your solution.
Answers
Answer: for one hard cover book it is about $17 and paperback is $13
Step-by-step explanation:
Two cars leave simultaneously from points A and B, the distance between which is 280 km. If the cars move to meet each other, they’ll meet in 2 hours. But if they move in the same direction, then the car going from point A will catch up with the car going from point B in 14 hours. What is the speed of each of the cars?
Answers
Answer:
Car A speed = 80 m/s
Car B speed = 60 m/s
Step-by-step explanation:
Let Speed of Car A be represented as x
Let Speed of Car B be represented as y
280 = (x+y)(2)140 = x+y --> Equation 1Let z represent the distance that Car A travels until it catches up to Car B
280+z = x(14)Since it takes 14 hours for Car B to cover the distance z,
z = 14y280+14y = 14x280 = 14(x-y)20 = x-y --> Equation 2Solving Equation 1 and Equation 2, we get:
x = y+20140 = 2y+20120 = 2yy = 60 x = y+20 = 80Answer:
Car A speed = 80 km/h
Car B speed = 60 km/h
Step-by-step explanation:
at the univariate level, descriptive statistics are used for: a. measures of association. b. data expansion. c. multivariate analyses. d. data reduction. e. determining causality.
Answers
The correct answer is at the univariate level, descriptive statistics are used for 'data reduction'
By creating the summaries of data samples, descriptive statistics can be used to describe the characteristics of a data set.
It is frequently presented as a summary of data that explains the contents of data. For instance, a census of the population may contain descriptive data on the proportion of men and women in a particular city.
Informational and intended to describe the precise features of a data set, descriptive statistics provide details. The greatest individual batting average for a player, the average number of runs per division, and the number of runs allowed per team are some descriptive statistics that may be used to analyse data from the previous Major League Baseball season.
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Find the sum.
-1/2 + 3/10
Answers
Answer:
-0.2 that is the answer
Answer:
-0.2,-1/5
Step-by-step explanation:
which function would be produced by a horizontal stretch of the graph of y = √x followed by a felection in the x-axis
Answers
Answer:
\(y=- \sqrt{\dfrac{1}{2}x}\)
Step-by-step explanation:
Parent functions are the simplest form of a given family of functions.
Transformations of graphs (functions) is the process by which a function is moved or resized to produce a variation of the original (parent) function.
Transformations
\(\begin{aligned} y=f(ax) \implies & f(x) \: \textsf{stretched/compressed horizontally by a factor of} \: a \\ & \textsf{If }a > 1 \textsf{ it is compressed by a factor of}\: a \\ & \textsf{If }0 < a < 1 \textsf{ it is stretched by a factor of}\: a \end{aligned}\)
\(y=-f(x) \implies f(x) \: \textsf{reflected in the} \: x \textsf{-axis}\)
Given parent function:
\(y=\sqrt{x}\)
Horizontal stretch:
As this is a horizontal stretch, the x variable should be multiplied by a value between zero and 1:
\(\implies y= \sqrt{\dfrac{1}{2}x}\)
Reflected in the x-axis:
To reflect a function in the x-axis, simply make the function negative:
\(\implies y=- \sqrt{\dfrac{1}{2}x}\)
The graph of a linear function is shown
Which word describes the slope of the line?
positive
negative
zero
undefined
Answers
Answer:
the answer is b
Step-by-step explanation:
plz give me a Brainliest
A word that describes the slope of the line is negative. The correct answer is option B.
What is a graph?A graph is the representation of the data on the vertical and horizontal coordinates so we can see the trend of the data. It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
For this case the first thing you should know is that we have an equation of the form:
y = mx + b
Where,
m: The slope of the line
b: Cutting point with the vertical axis.
For this case, we observe that the values of y decrease when the values of x increase. Therefore, the function decreases.
This means that it is true that:
m < 0
Therefore, the slope of the line is negative.
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10, 110, 210, 310, …
find a^30
Answers
The 30th term value is 2900.
What is an arithmetic sequence?A series of integers called an arithmetic progression or arithmetic sequence (AP) has a constant difference between the terms.
Given:
A sequence,
10, 110, 210, 310, …
To find the common difference:
110 - 10 = 100
So, d = 100
Remember that,
aₙ= a₁ + (n-1)d
aₙ=a₁+(n−1)d
So,
aₙ = 10 +(n−1)100.
a₃₀ = 10 + (30 - 1)100
a₃₀ = 10 + 29(100)
a₃₀ = 10 + 2900
a₃₀ = 2910.
Therefore, a₃₀ = 2910.
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There are red, blue and green counters in a bag in the ratio 2:5:3
What fraction of the counters are blue?
Answers
Answer:
5/10 or 1/2
Step-by-step explanation:
ratio=2:5:3
2+5+3=10 ( total of counters)
10=denominator
blue = ratio 5
5/10
or 1/2 simplified
A hot air balloon starts at an elevation of 300 feet. Then, it ascends at a rate of 600 feet per minute. what is the slope of the line?
Answers
Answer:
m = 600 feet/minute
Step-by-step explanation:
In this scenario, the elevation of the hot air balloon can be represented as a linear function of time. Let's use t to denote time in minutes and h(t) to denote the elevation of the balloon in feet at time t.
We know that the balloon starts at an elevation of 300 feet, so we can write the equation of the line as:
h(t) = 600t + 300
The slope of the line represents the rate of change of the elevation with respect to time, which is the same as the rate at which the balloon is ascending. Therefore, the slope of the line is equal to the ascent rate of the balloon, which is 600 feet per minute.
So the slope of the line is:
m = 600 feet/minute
Which of these numbers rounds to 7.62
7.546
7.639
7.670
7.720
HELP mE
Answers
Answer:
7.639
Step-by-step explanation:
7.639 rounds to 7.64 which is the closest on I could find to 7.62.
Answer:
I think the answer is 7.636
Step-by-step explanation:
HELPPPPPP MEEEEE PLEASEEEEEEEEE
Answers
Answer:6x-2y=8
Step-by-step explanation:
\(\left \{ {{3x-y=4} \atop {Ax-By=8}} \right. \\\frac{3x}{Ax} =\frac{y}{By} =\frac{4}{8}=\frac{1}{2}\\\frac{3}{A} =\frac{1}{2}\\A=6\\\frac{1}{B} =\frac{1}{2}\\B=2\\so, 6x-2y=8\)
Answer:
6x - 2y = 8
Step-by-step explanation:
If:
3x - y = 4
⇒ 4* 2 = 8
then:
3*2 = 6
-1 * 2 = -2
The answer is:
6x - 2y = 8
set up, but do not evaluate, an integral for the length of the curve. y = xex4, 0 ≤ x ≤ 8
Answers
The integral representing the length of the curve over the interval is as follows:
\(L=\int\limits^a_b {\sqrt{1+[e^(^x^4^)(1+4x^4)]^2} } \, dx\)
How to obtain the length of the curve?The length of a curve y = f(x) over an interval a ≤ x ≤ b is given by the following definite integral:
\(L=\int\limits^a_b {\sqrt{1+[f'(x)]^2} } \, dx\)
The interval is 0 ≤ x ≤ 8, hence the values of a and b are as follows:
a = 0, b = 8.
The curve is as follows:
y = xe^(x^4).
Applying the product rule, the derivative is given as follows:
y' = e^(x^4) + 4x^4e^(x^4)
y' = e^(x^4)(1 + 4x^4).
Hence the integral is as follows:
\(L=\int\limits^a_b {\sqrt{1+[e^(^x^4^)(1+4x^4)]^2} } \, dx\)
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What is the density of a substance that has a mass of 500 g and a volume of 500mL
Answers
Answer:
1
Step-by-step explanation:
Given,
Mass ( m ) = 500 g
Volume ( V ) = 500 ml
To find : Density ( D ) = ?
Formula : -
D = m / V
D = 500 / 500
D = 1 g/ml
Hence,
1 g/ml is the density of a substance that has a mass of 500 g and a volume of 500mL.
What is the length of BC?
Answers
Answer:
2.1
Step-by-step explanation:
This involves trigonometry. Since we need to find the value of a, we need to find a trig equation with a as the numerator. Luckily, we have one, tan. So, tan(20) is equal to a/6, right. So, using a calculator, tan(20) is approximately equal to 0.363. This times 6 should equal a. This is approximately around 2.1. Thus our answer is 2.1
the number, , of people who have heard a rumor spread by mass media at time, , is given by there are 400000 people in the population who hear the rumor eventually. 15 percent of them heard it on the first day. find and , assuming is measured in days.
Answers
The number of people who heard the rumor on the first day is 60000.
To find the number of people, n, who have heard a rumor spread by mass media at time t, we can use the equation
\(n = 400000 * (1 - 0.85^t)\)
Where:
- n is the number of people who have heard the rumor at time t
- 400000 is the total population who eventually hear the rumor
- 0.85 is the percentage of people who didn't hear the rumor on the first day (1 - 0.15)
- t is the time in days
To find the number of people who heard the rumor on the first day, we substitute t = 1 into the equation:
\(n = 400000 * (1 - 0.85^1)\)
n = 400000 * (1 - 0.85)
n = 400000 * 0.15
n = 60000
Therefore, the number of people who heard the rumor on the first day is 60000.
In conclusion, the answer is that 60000 people heard the rumor on the first day.
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Write an equation, in standard form, for the line that passes through the points (-1, -4) and (1, 6).
Answers
Answer:
y = 7/2 x + 5/2Step-by-step explanation:
The equation of a line is expressed as y = mx+c where m is the slope and c is the intercept.
m = Δy/Δx = y₂-y₁/x₂-x₁
Given two points (-1, -4) and (1, 6) with x₁= -1, x₂ = 1, y₁ = -4 and y₂ = 6
m = 6-(-1)/1-(-1)
m = 7/2
To get c, we will substoute the slope and any of the point given into the equation of a line as shown;
Using tyhe point (1, 6) and m = 7/2
6 = 7/2(1)+c
6 = 7/2 + c
c = 6-7/2
c = 5/2
The equation of the line will be y = 7/2 x + 5/2
Help me please yall id appreciate it
Answers
Answer:
∠ IMN = 70°
Step-by-step explanation:
∠ KIM = ∠ JIL ( vertically opposite angles ), that is
x + y = 102 → (1)
∠ IMN = 3x - 26 ( vertically opposite angles ) , that is
y = 3x - 26 ← substitute into (1)
x + 3x - 26 = 102
4x - 26 = 102 ( add 26 to both sides )
4x = 128 ( divide both sides by 4 )
x = 32
Then
∠ IMN = y = 3x - 26 = 3(32) - 26 = 96 - 26 = 70°
Answer:
m∠IMN = 70°--------------------------
Set up equations for the vertical angles:
x + y = 1023x - 26 = yRewrite the second equation:
3x - y = 26Add up this with the first equation:
x + y + 3x - y = 102 + 264x = 128x = 128/4x = 32Find y:
y + 32 = 102y = 102 - 32y = 70The angle IMN is same as y, so its measure is 70°.
Rewrite in simplest terms:(x+9y)-(9x-8y)
Answers
The simplified expression is (-8x + 17y).
We have,
(x+ 9y) - (9x - 8y)
Now, simplifying the given expression as
(x+ 9y) - (9x - 8y)
= x + 9y - 9x + 8y
= x - 9x + 9y + 8y
= -8x + 17y
Thus, the simplified expression is (-8x + 17y)
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34x +7y - 45x + 2y - 12x
Answers
Combining like terms: -23x + 9y
I don’t know if you need it simplified or not but here is the simplified form
It takes 120 minutes to drive home at 60kmh
how long it will take to drive home at 120kmh
Answers
Answer: 240 mins
Step-by-step explanation:
Answer = 60mins
Select the correct answer.
Which number best represents the slope of the graphed line?
A. -5
B.-1/5
C. 1/5
D.5
Answers
Answer:
A. -5
Step-by-step explanation:
You want the slope of a graphed line that crosses grid points (0, 2) and (1, -3).
SlopeThe sign of the slope is positive if the line goes up to the right. This line goes down to the right, so the sign of its slope is negative. (Eliminates choices C and D)
The magnitude of the slope is the ratio of vertical distance to horizontal distance between two points. This line drops 5 vertical squares for each square to the right, so its slope is ...
rise/run = -5/1 = -5
The slope is best represented by -5.
__
Additional comment
From a point on the line that is a crossing of grid lines, if the line crosses the next vertical grid line before it crosses the next horizontal grid line, the magnitude of its slope is less than 1.
Here, the line crosses several horizontal grid lines before it crosses another vertical grid line, so its slope has a magnitude greater than 1. When the choice is between -5 and -1/5, the larger magnitude number is -5.
From two points on the line, you can find the slope using the formula ...
m = (y2 -y1)/(x2 -x1)
m = (-3 -2)/(1 -0) = -5/1 = -5 . . . . . . using points (0, 2) and (1, -3)
A shop sells
17 different types of sandwich
15 different types of drink
26 different types of snack
14 different types of dessert.
A meal deal consists of either
a sandwich, a drink and a snack
or a sandwich, a drink and a dessert.
(a) Show that there are over 10000 different ways to choose a meal deal
Answers
If a shop sells 17 different types of sandwich,15 different types of drink, 26 different types of snack, 14 different types of dessert. The number of different ways is 10,200.
How to find the total number of different ways?If a meal deal consists of a sandwich, a drink and a snack
Number of different ways = 17 × 15 × 26
Number of different ways = 6,630 ways
If a meal deal consists of sandwich, a drink and a dessert
Number of different ways = 17 × 15 ×14
Number of different ways = 3,570
Total number of different ways :
Total number of different ways = 6,630 ways + 3,570 ways
Total number of different ways =10,200 ways
Therefore we can conclude that there are over 10,000 different ways to choose a meal deal as the total number of different ways is 10,200 ways.
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i cant figure out the answer
Answers
Answer: 3/10
Step-by-step explanation: times
Answer:
18
Step-by-step explanation:
\(\frac{1}{5}\) × 60 = 12 ← number who chose basketball
\(\frac{1}{2}\) × 60 = 30 ← number who chose volleyball
That is a total of 12 + 30 = 42 students
Since the remaining students chose soccer, then
60 - 42 = 18 ← number who chose soccer
what do these equal
A=
B=
C=
D=
Answers
Some units are considered for Maximum Time Frame calculations, and others aren't. Which of these is correct? a. If you take an Incomplete or Withdrawal this will not be included. B. F grades won't count toward your Max Time Frame. C. When you withdraw from a class, it's taken out of the calculation for Max Time Frame. D. If you drop a class before the dropladd deadline it will not show up as an attempted class
Answers
All of the given options are generally correct regarding the considerations for Maximum Time Frame calculations.
Let's break down each option:
a. If you take an Incomplete or Withdrawal, this will not be included: In most cases, if you receive an Incomplete or withdraw from a course, the units for that particular course will not be included in the calculation for Maximum Time Frame.
b. F grades won't count toward your Max Time Frame: Usually, failing grades (F) are not counted towards the Maximum Time Frame calculation since they do not contribute to earned credits.
c. When you withdraw from a class, it's taken out of the calculation for Max Time Frame: Generally, when you withdraw from a class, the units for that class are not considered in the calculation for Maximum Time Frame.
d. If you drop a class before the drop/add deadline, it will not show up as an attempted class: Typically, if you drop a class before the drop/add deadline, it will not be considered as an attempted class and therefore will not be included in the Maximum Time Frame calculation.
However, it's important to note that policies regarding Maximum Time Frame calculations may vary among educational institutions. It's always advisable to consult your institution's specific policies or contact the appropriate academic advisors for accurate information regarding Maximum Time Frame calculations at your institution.
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Please help fast Prefer A Expert or Above to Help Me out it’s the bottom one!
Answers
The value is approximate.
==========================================================
Explanation:
Focus on the upper triangle.
Label the unknown horizontal side as y
Use the pythagorean theorem to solve for y
a^2+b^2 = c^2
10^2 + y^2 = 26^2
100+y^2 = 676
y^2 = 676-100
y^2 = 576
y = sqrt(576)
y = 24
-----------
Now move to the bottom triangle
With respect to the reference angle, the side y = 24 is opposite this. The hypotenuse is unknown.
Use the sine rule to connect the two and solve for x
sin(angle) = opposite/hypotenuse
sin(32) = y/x
sin(32) = 24/x
x*sin(32) = 24
x = 24/sin(32)
x = 45.2899179551967 which is approximate
x = 45 rounding to the nearest whole number
Side x is approximately 45 cm long.
Note: make sure your calculator is in degree mode.
6. if p(a∪b)=0.8, p(a∩b')=0.3, and p(b∩a')=0.2, determine following probabilities. 6-a. p(a∪b)' 6-b. p(a∩b) 6-c. p(a) 6-d. p(b) 6-e. p(b')
Answers
Probability is a measure of the likelihood of an event to occur. If p(a∪b)=0.8, p(a∩b')=0.3, and p(b∩a')=0.2 then 6-a. p(a∪b)'=0.2, 6-b. p(a∩b) = 0, 6-c. p(a) = 0.3 6-d. p(b) =0.2, 6-e. p(b') =0.8.
Using De Morgan's laws and the formula for the probability of the union of two events to solve these problems.
Let A' denote the complement of event A.
6-a. We have:
p(A∪B)' = 1 - p(A∪B)
Since A and B are not necessarily disjoint, we can use the formula:
p(A∪B) = p(A) + p(B) - p(A∩B)
Substituting the given values, we get:
p(A∪B) = p(A) + p(B) - 0.3 = 0.8
We don't know the values of p(A) and p(B) individually, so we can't calculate p(A∪B) directly. However, we can use the formula for the complement and the values we already have:
p(A∪B)' = 1 - p(A∪B) = 1 - 0.8 = 0.2
Therefore, p(A∪B)' = 0.2.
6-b. We are given that p(A∩B') = 0.3. Using De Morgan's laws, we can rewrite this as:
p((A')∪B) = 0.3
Now, we can use the formula for the probability of the union of two events to get:
p(A') + p(B) - p(A'∩B) = 0.3
Since A and B are linearly independent, we know that A' and B are also linearly independent. Therefore, A'∩B = {0}, the zero vector. So we have:
p(A') + p(B) = 0.3
We also know that p(B∩A') = 0.2. Again using De Morgan's laws, we can rewrite this as:
p((B')∪A) = 0.2
Using the formula for the probability of the union, we get:
p(B') + p(A) - p(B'∩A) = 0.2
Since A and B are linearly independent, we know that B' and A are also linearly independent. Therefore, B'∩A = {0}. So we have:
p(B') + p(A) = 0.2
Now we have two equations with two unknowns. Solving for p(A) and p(B), we get:
p(A) = 0.4
p(B) = 0.1
Therefore, p(A∩B) = p({0}) = 0.
6-c. We have:
p(A) = p(A∩B) + p(A∩B') = 0 + 0.3 = 0.3
Therefore, p(A) = 0.3.
6-d. We have:
p(B) = p(A∩B) + p(B∩A') = 0 + 0.2 = 0.2
Therefore, p(B) = 0.2.
6-e. We have:
p(B') = 1 - p(B) = 1 - 0.2 = 0.8
Therefore, p(B') = 0.8.
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If yx=lnx then find dy/dx Select one: a. xlnx1−lny b. xlnx1 C. x2yxy−xylny d. xylny e. x2lnxy
Answers
The derivative of y to x is dy/dx = 1/(xe^y). Hence, the option e. x2lnxy is the correct choice.
Given the relation yx = ln x.
To find: dy/dx
We can differentiate both sides of the equation yx = ln x to x as follows:
(yx)' = (ln x)'
Let us now apply the chain rule of differentiation:
Product of two functions:
yx = ln x; we can rewrite it as y = ln u, where u = x and hence
x = eu(yx)' = (ln u)'(u)'
Using the differentiation formula of the natural logarithm function
(ln u)' = 1/u, we have (yx)' = (ln u)'/u
Since u = x and x = eu, we have
(ln u)' = 1/u = 1/x, giving
(yx)' = (ln u)'/u
= (1/x) (1/eu)
= 1/(xeu)
To summarize, dy/dx = 1/(xeu). Thus, the correct option is e. x2lnxy.
Further simplification leads to (yx)' = 1/(xeu) = 1/(xe^y). Thus, the derivative of y to x is dy/dx = 1/(xe^y). Hence, the option e. x2lnxy is the correct choice.
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