Show f(x)=x^3+e^x has exactly one real root?

Answers

Answer 1

Strategy: We need to do two things here. First, we show that the equation has a solution. Then we will need to show that there cannot be two solutions. We will use the Intermediate Value Theorem for the existence of a solution, and Rolle’s Theorem (or the MVT) for the fact that there is only one. Our strategy will include the idea that for the function f(x) = x3 + ex, this problem is equivalent to saying that the graph of f(x) has only one x-intercept.
Solution: First note that the function f (x) = x3 + ex is differentiable on all of R, since it is the sum of two differentiable functions, a polynomial and the exponential function. Hence it is continuous. Grabbingthetwoinputvaluesx=−1andx=0,weseethatf(−1)=(−1)3+e−1 =−1+1e <0, and f (0) = (0)3 + e0 = 1 > 0. Hence by the Intermediate Value Theorem, we know that there must be a number c ∈ [−1, 0], where f (c) = 0. But this is our solution to the original equation. Hence a solution exists.
So call c our solution to the original equation and suppose there exists another one; Call this one d. Then either d > c or d < c. It really won’t matter for our discussion, so assume d > c. Then f(x) is a continuous function on [c,d] and differentiable on (c,d) (it is differentiable everywhere, actually). By Rolle’s Theorem, there then must exist a point r ∈ (c,d), where f′(r) = 0. However, we have
f′(x) = 3x2 + ex > 0
for all x ∈ R. Thus, like our example in class, there cannot be a point r ∈ (c,d), where f′(r) = 0. So our assumption that there is a second solution to the original equation is incorrect. Thus there is only one real solution to x3 + ex = 0.

Related Questions

Which value must be added to the expressionx2+16x to make it a perfect-square trinomial?

Answers

Answer:

Step-by-step explanation:

Please write x^2+16x or x²+16x to indicate the exponentiation.

To rewrite x²+16x as a perfect square trinomial:

1) Take half of the coefficient of x, that is, half of 16, which is 8

2) Square this result: 8² or 64

3) Starting with x²+16x again, add 64 and then subtract 64 to get a trinomial expression equivalent to x²+16x:

x²+16x + 64 - 64

4) Rewrite the first three terms as the square of the binomial x + 8:

(x + 8)² - 64

Here (x + 8)² is a perfect square trinomial.

Answer:

The answer is C.) 64 on edge.

Is 2x+y=4 linear? If it is what is the slope and y intercept?

Answers

Answer: Yes, it is linear. The slope is -2

Step-by-step explanation:

move the 2x over to the other side and get y=-2x+4, the slope in a linear equation is m in the formula y=mx+b, in this case, m=-2

To rent a Bird scooter you must pay $1 to start and then $0.50 for every 15 minutes you ride. Is the cost proportional to the amount of time you ride?

Answers

Step-by-step explanation:

if you rent a scooter for 15 minutes you pay $1.50 ($1 to start and $0.50 rental) and if you rent it for 30 minutes then you pay $2.00 ($1 to start and $1 rental). This is NOT proportional because $1.50/15-mins and $2.00/30-min are NOT equivalent ratios

1/3 - 2/9 find the solution

Answers

Answer:

1/9

Step-by-step explanation:

1/3 - 2/9

Get a common denominator of 9

1/3 *3/3 = 3/9

3/9 - 2/9

1/9

Can someone help me with 5 and 6 please

Answers

Answer:

5. x+4=86

6. An equation can be used to predict values from a pattern by merely substituting any value to determine the equivalent value. An example is the geometric series which has a standard equation of an = a1 * r^(n-1) and is used to predict the nth term of the series

Step-by-step explanation:

The legs of a right triangle are 10 cm and 24 cm long what is the length of the hypotenuse

Answers

Answer:

Step-by-step explanation:

the formula for this is A squared plus B squared equals C squared

Answer:

Step-by-step explanation:

10(10)+(24)(24)

100+576

676

out of the square root

it is 26

Tisha orders a carpet for a room that measures 270 square feet. The salesman says the carpet costs $12.00 per square yard. He explains that since there are 3 square feet for every square yard, tisha needs 90 square yards of carpeting, which cost $1,080. What error did the salesman make? What should be the actual cost of the carpet?

Answers

Answer:

What error did the salesman make?

The salesman made an error when convert 270 square feet to yards.

What should be the actual cost of the carpet?

The actual cost of the carpet is $360

Step-by-step explanation:

From the question, we are told that:Tisha orders a carpet for a room that measures 270 square feet.

This means the room needs 270 square feet of carpet

But from the question, we know that the carpet is sold in square yards

We convert square feet to square yards

For our conversion rates

3 ft = 1 yard

9 ft² = 1 square yards(yd²)

270 ft² = x

Cross Multiply

9 ft² × x = 270ft² × 1 yd²

x yd²= 270ft² × 1 yd²/9 ft²

x = 30 yd²

Hence, 270ft² = 30yd²

Cost of carpet =

1 yd² = $12.00

30 yd² =

Cross Multiply

= 30 × $12.00

= $360.00

The actual cost of the carpet is $360

What error did the salesman make?

The salesman made an error when convert 270 square feet to square yards.

He said 270 square feet = 90 square yards

But he was wrong.

270 square feet = 30 square yards.

What should be the actual cost of the carpet?

The actual cost of the carpet is $360

Answer:360 for the actual cost

Step-by-step explanation:

Jasmine drives every morning to her office that is 12 miles from her home.

If Jasmine's average driving speed is s mph, how many hours does it take her to drive from her home to the office?

Answers

Answer:

is it s/12

Step-by-step explanation:

How can you use Recipes B and D to create recipe E?

Fill in the blank.

1.) ____ the corresponding quantities of Recipe B 2.)____ Recipe D to create Recipe E.

Options for 1.)

- Add
- Subtract
- Multiply
- Divide

Options for 2.)

- and
- from
- times
- by

Answers

1.) add
2.) and

If you look at the chart you can see when you add B and D it equals E, therefore your answer is add.

i need help with 1/2x+13=9
im in the 8th grade btw

Answers

Answer:

-8

Step-by-step explanation:

You are wanting to solve for x

First you want to subtract 13 on both sides-

1/2x = -4

Now you want to divide on both sides-

x = -8

Answer:

1/-8

To find:

Step-by-step explanation:

1/2x+13=9

take 13 to the LHS. when you take 13 to the LHS it becomes negative

1/2x=9-13

1/2x=-4

by cross multiplying,

2x × -4=1

-8x=1

x=1/-8

hope it helps you!!

A LOT OF POINTS!!

find the value of x for each figure. each figure is in centimeters(cm)

please answer this i have been giving away points but nobody answers :(

Answers

Answer:

x*(6/10)=6/70

x=6/70*10/6

x=1/7

x*2/8=6/32

x=6/32*8/2

x=3/4

x*1/5=1/15

x=1/15*5=1/3

x*1/7=1/14

x=1/14*7=1/2

x*1/2=1/4

x=1/4*2=1/2

x*2/5=14/45

x=14/45*5/2

x=7/9

which one is it??.......

Answers

it is rational so the 2nd one

James decides to invest money in Wal-Mart. As a shareholder, he has the advantage of

(select all that apply)

A. limited liability

B. limited life

C. unlimited liability

D. double taxation

E. ease of start up

Answers

Answer:

A. limited liability

Step-by-step explanation:

Because James has become a shareholder by investing in Walmart, he has the advantage of limited liability. This basically means that he shares in both the gains and the losses of Walmart's business decisions but only up to the amount that he has invested into it. Meaning that if James invested a total of $2000 and Walmart goes under, then he will lose all $2000 but not more than that.

A bag with 2 ½ ounces of peanuts can make ⅔ of a jar of peanut butter.

How many ounces of peanuts will it take for there to be one jar of peanut butter?

Answers

Given :

A bag with 2 ½ ounces of peanuts can make ⅔ of a jar of peanut butter.

To Find :

How many ounces of peanuts will it take for there to be one jar of peanut butter.

Solution :

[tex]2\dfrac{1}{2}\ ounces = \dfrac{2}{3}\ jar\\\\\dfrac{5}{2}\ ounces = \dfrac{2}{3}\ jar\\\\1= \dfrac{15}{4}\ \dfrac{ounces}{jar}[/tex]

Now,

1 jar of peanut butter is made by :

[tex]1\times jar \times \dfrac{15}{4} \dfrac{ounces}{jar}\\\\\dfrac{15}{4}\ ounces[/tex]

Hence, this is the required solution.

The radius of a circle is 8 meters. What is the diameter?

Answers

Answer:

16 meters

Step-by-step explanation:

r=8m

8*2=16

d=16m

Answer:

Step-by-step explanation:

The diameter is the radius times 2.

Hi, i need help quickly. I need the solution for
Estimate the value of
[tex] \sqrt{97.5 \div 1.96} [/tex]
Please make it quick

Answers

Answer:

7.05

Step-by-step explanation:

Answer:

7.053

Step-by-step explanation:

97.5/1.96=49.745

√49.745=7.053

HELP ASAP . You the real one if you do.

Answers

Answer:

x=1

Step-by-step explanation:

hope this helps and hope its correct

does the table below show a proportional relationship

Answers

Answer:yes

Step-by-step explanation:

need help really bad please

Answers

Problem 1

Answer: Choice B) angle1 = 26, angle2 = 64

----------

Explanation:

Complementary angles add to 90

(angle1)+(angle2) = 90

(5x+6)+(68-x) = 90

4x+74 = 90

4x = 90-74

4x = 16

x = 16/4

x = 4

So,

angle1 = 5x+6 = 5(4)+6 = 26

angle2 = 68-x = 68-4 = 64

Note how

angle1+angle2 = 26+64 = 90

to help check our answer

==========================================

Problem 2

Answer: Choice D) BC, AC, AB

----------

Explanation:

The shortest side of a triangle is always opposite the smallest angle.

The longest side of a triangle is always opposite the largest angle.

Based on those two facts, we can say the shortest side is BC (since angle A = 53 is the smallest angle) and the longest side is AB (since angle C = 70 is the largest)

The order from smallest to largest is

BC, AC, AB

==========================================

Problem 3

Answer: Choice D) B < C < A

----------

Explanation:

Use the reverse of the idea from problem 2. We know that AC is the smallest side which must mean that angle B is the smallest angle. Since BC is the largest side, angle A must be the largest angle.

The order of the angles is B < C < A

==========================================

Problem 4

Answer: Choice A) angle 5 = 50, angle 8 = 130

----------

Explanation:

The same side exterior angles must be supplementary to have parallel lines.

==========================================

Problem 5

Answer: Choice A) angle G is the smallest

----------

Explanation:

Use the same idea as problem 2. The smallest angle is opposite the smallest side. The smallest side is 48, so angle G being opposite of that side, is the smallest angle.

Angle O would be the largest as it is opposite the largest side DG = 70

Solve for the indicated variable.

2a + 2b = c for b

B = ?/?

Answers

Answer:

b = c − 2a

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

Step-by-step explanation:

2a + 2b = c

2b = c − 2a

b = c − 2a

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

multiply this. 5 • (-3)

Answers

Your answer is going to be -15

What is the slope of this line??

Answers

-6/7 because it’s rise over run, we are going down so it’s negative, 10 down and 9 to the left which is -0.9 and -6/7 it’s -0.857 and if we round this number we have -0.9 left

-4.5x(-2) =
i need help solving this equation

Answers

Answer:

Step-by-step explanation:

-4.5X(-2)

-4.5x * -2

Write an equation in slope-intercept form of the line that passes through (6,-1) and (3,-7). Please help!

Answers

Answer:

y = 2x – 13

Step-by-step explanation:

Use Slope formula to get the slope

(y2-y1) / (x2-x1)

(-7--1) / (3-6)

(-6)/(-3)

Then plug-in the value of slope and any one coordinate in Point-Slope formula:

y- y1= m(x-x1)

y - -1 = 2( x - 6)

y + 1 = 2x - 12

y = 2x - 13

The equation of the line in the slope-intercept form will be y = 2x – 13.

What is an equation of the line?

An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.

The general form of the equation of the line:-

y = mx + c

m = slope

c = y-intercept

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope = (-7-(-1)) / (3-6)

Slope = (-6)/(-3)

Slope = 2

Then plug in the value of slope and any one coordinate in the Point-Slope formula:

y- y₁= m(x-x₁)

y - (-1) = 2( x - 6)

y + 1 = 2x - 12

y = 2x - 13

Therefore, the equation of the line in the slope-intercept form will be y = 2x – 13.

To know more about an equation of the line follow

brainly.com/question/18831322

#SPJ2

A gym offers kickboxing classes 22 of the 30 days in October. What decimal is equivalent to the fraction of days in October that classes were offered at the gym?

Answers

0.290332258 i’m not sure what you want it rounded to.

explaination:
figure out how many days that is (9) then divide that by how many days in october (31)

What is the volume of the toy car that has a density of 3 g/mL and a mass of 75 g.

Answers

225 mL

75/x = 3/1 g/mL

Answer:

25 mL

Step-by-step explanation:

m/D=V

75g/3 g/mL = V

25mL =V

3. Three meals per day are provided by the hotel. The hotel will charge a total of $9,240 for all meals for the entire group. What is the cost of meals for each person?

Answers

Answer:

1 meal = $3.080

Joe had a summer job that pays $7.00 an hour and he worked 15 and 35 hours every week. His weekly salary can be modeled by the equation: s = 7H, wear a is his weekly salary and h is the number of hours he worked in a week. Describe the domain and range for this problem using appropriate notation.

Answers

Correct question is;

Joe had a summer job that pays $7.00 an hour and he worked between 15 and 35 hours every week. His weekly salary can be modeled by the equation: S = 7h, where S is his weekly salary and h is the number of hours he worked in a week.

What is the reasonable Domain and Range?

Answer:

Domain is: 15 ≤ x ≤ 35

Range is: 105 ≤ x ≤ 245

Step-by-step explanation:

We are told that Joe works between 15 and 35 hours every week.

This means that he works at least 15 hours or at most 35 hours a week.

Thus, domain is; 15 ≤ x ≤ 35

We are told that his weekly salary can be modeled by the equation: s = 7H

Where s is salary and H is number of hours worked per day.

Thus;

Lower range is; s = 7 × 15 = $105

While higher range is; s = 7 × 35 = $245

Thus, range is 105 ≤ x ≤ 245

Help plsss due today!!!

Answers

Answer:

320 feet

Step-by-step explanation:

your mom

hope this helps:)

Can someone please help me ASAP I’ll mark brainlist!!!!?!?

Answers

It might be D. I’m not positive sorry if it’s wrong

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