Consider the function below. (If an answer does not exist, enter DNE.) f(x) = 1 2 x4 − 4x2 + 5 (a) Find the interval of increase. (Enter your answer using interval notation.) Find the interval of decrease. (Enter your answer using interval notation.) (b) Find the local minimum value(s). (Enter your answers as a comma-separated list.) Find the local maximum value(s). (Enter your answers as a comma-separated list.) (c) Find the inflection points.

Answers

Answer 1

Answer:

Please, read the answer below.

Step-by-step explanation:

You have the following function:

[tex]f(x)=12x^4-4x^2+5[/tex]

(a) To find the intervals of increase or decrease of f(x) you first calculate the derivative of f(x):

[tex]\frac{df}{dx}=\frac{d}{dx}[12x^4-4x^2+5]\\\\\frac{df}{dx}=12(4)x^3-4(2)x=48x^3-8x[/tex]   (2)

Next, you equal the derivative to zero and obtain the roots of the obtained polynomial:

[tex]48x^3-8x=0\\\\6x^3-x=0\\\\x(6x^2-1)=0[/tex]

Then, you have the following roots for x:

[tex]x_1=0\\\\x_{2,3}=\pm \sqrt{\frac{1}{6}}[/tex] =  ±0.40

Hence, there are three special points.

Next, you evaluate the derivative (expression (2)) for the x values close to the x1, x2 and x3. The values of the derivative give to us the value of the slope of a tangent line in that point, and so, if the function increases or decreases:

First interval, for a number lower than -0.40

[tex]48(-0.41)^3-8(-0.41)=-0.02<0[/tex]

The function decreases in the interval:

[tex](-\inft,-\frac{1}{\sqrt{6}})[/tex]

It is necessary that after x=-0.40 the function increases until the next special point, that is x=0. Then, the interval in which the function increases is:

[tex](-\frac{1}{\sqrt{6}},0)[/tex]

By symmetry, from the point x=0 until x=0.40 the function decreases.

[tex](0,\frac{1}{\sqrt{6}})[/tex]

Next, you evaluate the expression (2) for a number higher than 0.40:

[tex]48(0.41)^3-8(0.41)=0.02>0[/tex]

Then, the function increases for the following interval:

[tex](\frac{1}{\sqrt{6}},+\infty)[/tex]

(b) Due to the results obtained in the previous step you can conclude that the local minimum are:

[tex]x_{min}=-\frac{1}{\sqrt{6}}\\\\x_{min}=\frac{1}{\sqrt{6}}[/tex]

[tex]P_1(-\frac{1}{\sqrt{6}},f(-\frac{1}{\sqrt{6}}))=P_1(-\frac{1}{\sqrt{6}},4.66)\\\\P_2(\frac{1}{\sqrt{6}},f(\frac{1}{\sqrt{6}}))=P_2(\frac{1}{\sqrt{6}},4.66)[/tex]

[tex]P_1(-\frac{1}{\sqrt{6}},0)\\\\P_2(\frac{1}{\sqrt{6}},0)[/tex]

(these are the point in which the function change of a decrease to an increase)

The same reason as before. There in one local maximum:

[tex]x_{max}=0[/tex]

[tex]P(0,f(0))=P(0,5)[/tex]

(c) The inflection points are calculated by using the second derivative:

[tex]\frac{d^2f}{dx^2}=144x^2-8=0\\\\x^2=\frac{8}{144}=\frac{1}{18}\\\\x=\pm \frac{1}{\sqrt{18}}[/tex]

Then , there are two inflection points , given by:

[tex]P_1(-\frac{1}{\sqrt{18}},f(-\frac{1}{\sqrt{18}}))=P_1(-\frac{1}{\sqrt{18}},4.82)\\\\P_2(\frac{1}{\sqrt{18}},f(\frac{1}{\sqrt{18}}))=P_2(\frac{1}{\sqrt{18}},4.82)[/tex]


Related Questions

What is the classification for this polynomial?
7
Click on the correct answer.
monomial
binomial
trinomial

Answers

Answer:

7 is monomial

Step-by-step explanation:

An expression with only one term is called a monomial

✌️;)

7 !.!!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.

Can someone answer I barely have any points now :(

Answers

A. True, a rhombus is a parallelogram with equal sides. A rhombus has four congruent sides.

B. True, but there are two other ways to find the area of a rhombus: there is the "diagonals" method and the "trigonometry" method.

C. False, the area of a rhombus is not less than the area of a parallelogram because it will depend on the diagonals.

D. False, a parallelogram is not always a rhombus because the dimensions and other attributes of a parallelogram may vary while the rhombus will remain equal for all aspects.

E. True, if you use the base times height formula but if you used any of the other formulas this would be false.

What is the sixth term in the sequence 3, 6, 11, 18

Answers

Answer:

38

Step-by-step explanation:

3 to 6 is an add by 3 then 6 to 11 is an add by 5 then 11 to 18 is an add by 7 each add is increased by 2 so the next add will be 9 giving you 27 then you will add 11 because it's increased by two giving you 38

Trina has a total of 32 crayons. How much boxes of crayons did she buy.

Answers

Answer:

Depends on the type of box, but most likely one box of crayons because 32 packs are sold.

Do you have answer choices?

The height of a triangle is 18 inches. Find the length of the base if the area of the triangle is 63 square inches

Answers

Answer:

7 inches

Step-by-step explanation:

The area of a triangle is found by

A = 1/2 bh

We know the height is 18 and the area is 63

63 = 1/2 (b)(18)

63 = 9b

Divide each side by 9

63/9 = 9b/9

7 = b

The base is 7 inches

Answer:

7 inches

Step-by-step explanation:

Area = ½ base × height

63 = ½ × base × 18

63 = 9 × base

base = 7 inches

What point appears to be the solution to the system of
equations shown in the graph?

Answers

Answer:

x = -2

y= -5

is the solution of the system of equations

To generate leads for new business, Gustin Investment Services offers free financial planning seminars at major hotels in Southwest Florida. Gustin conducts seminars for groups of 25 individuals. Each seminar costs Gustin $3,500, and the commission for each new account opened is $5,000. Gustin estimates that for each individual attending the seminar, there is a 0.01 probability that he/she will open a new account.

(a) Determine the equation for computing Gustin's profit per seminar, given values of the relevant parameters. Profit = (New Accounts Opened × ) –

(b) What type of random variable is the number of new accounts opened? (Hint: Review Appendix 11.1 for descriptions of various types of probability distributions.)

(c) Choose the appropriate spreadsheet simulation model to analyze the profitability of Gustin's seminars. (I) (II) (III) (IV) Would you recommend that Gustin continue running the seminars?

(d) How many attendees (in a multiple of five, i.e., 25, 30, 35, . . .) does Gustin need before a seminar's average expected profit is greater than zero?

Answers

Answer:

a) profit = (new account opened x 5000) -3500

b) Opening account is binomial distribution with n =25 and p = 0.01

c) Probability of loss is 0.77781 --I don't recommend the company that it running the seminar

d) n ≅ 71

Step-by-step explanation:

See attached image

Find the slope, if it exists, of the line containing the points (-1,-15) and (-17,-20)
The slope is _____________________

Answers

y=.3125x - 14.6875 or y=5/16x - 235/16


How many feet would a 27-inch object measure?

Answers

2 1/4 feet
2ft 3in
_______

Answer:

2 feet 3 inches

Step-by-step explanation:

So we know there are 12 inches in one foot. so 27/3= 2 remainder 3. So the answer is 2 feet 3 inches

The number 65 is decreased to 62. What is the percentage by which the number was decreased, to the nearest tenth of a percent?

Answers

Answer:4.6

Step-by-step explanation:

Percentage decrease=(65-62)/65 x 100

Percentage decrease=3/65 x 100

Percentage decrease=(3x100)/65

Percentage decrease=300/65

percentage decrease=4.6

What is the median of the following set of data ?



56.2 112.8 90.4 95.5 103.7 87.6 99.1 112.4 59.7

Answers

Answer: 95.5

Step-by-step explanation:

If you order the numbers from least to greatest and alternate revoking numbers the number in the middle, or median will be 95.5

Please help ASAP! I will mark BRAINLIEST! Please answer CORRECTLY! No guessing! CHECK ALL THAT APPLY

Answers

Answer:

D. 96

F. 32

Step-by-step explanation:

The inequality 3x ≥ 96, means that x and 3 must be multiplied to get a product that is greater than 96.  So to figure this out, you must plug in the numbers given for x, like so:

3x ≥ 96

A. 3(31) ≥ 96

     93 ≥ 96                            This is incorrect since 93 is not greater than 96

A tip that makes this easier is that, you can see that 31 multiplied by 3 is 93, which is only a couple numbers off 96, so all numbers after 31 will result in a product greater than 96. For example:

D. 3(96) ≥ 96

    288 ≥ 96                             This inequality is correct

F. 3(32) ≥ 96

   96 ≥ 96                        

The inequality sign states that the number must be either greater or equal to 96, which it is.

5×60=5×___tens
=_____tens
=______

Answers

Answer:

300

Step-by-step explanation:

5x60=5x6 tens it equal to 30 tens. 30 tens equals 300

Answer:

see below

Step-by-step explanation:

there are 6 tens in 60!

5x60 = 5 x 6 tens

= 30 tens

= 300

Solve (x + 6)2 = 64.

Answers

Answer:

x=26

Step-by-step explanation:

(x+6)2=64

x+6=32

x=26

Answer:

Here is the solution. Mark as brainlist please.

What is the volume of a right circular cylinder with a radius of 5 cm and a height of 12 cm?
Unit Te
4.05 Unit
EN OP
60 cm
120 cm
3007 cm
12007 cm

Answers

Answer:742 cm^3

Step-by-step explanation:

Height=h=12cm

Radius=r=5cm

π=3.14

volume of cylinder=π x r^2 x h

Volume of cylinder=π x r x r x h

Volume of cylinder=3.14 x 5 x 5 x 12

Volume of cylinder=742 cm^3

29. Find the circumference.
11
a. 3.14
b. 69.08
c. 121

Answers

Answer:

b.69.08

Step-by-step explanation:

formula:

c=2(pi)r

c=2(pi)11

Answer:b

Step-by-step explanation:

radius=r=11

π=3.14

Circumference=2 x π x r

Circumference=2 x 3.14 x 11

Circumference=69.08

What if f(x)=(1/7)(7x), what is f(3)

Answers

Answer:

  3

Step-by-step explanation:

f(x) = (1/7)(7x) can be simplified to ...

f(x) = x

Then for x=3, ...

  f(3) = 3

Ellis is 6 feet tall. If she currently has a 4 foot shadow, what angle are the sunrays making with the ground? Round decimal to the nearest hundredth.

Answers

Answer:

19.70

Step-by-step explanation:

For this case we have the following trigonometric relationship:

Tan (x) = (4/6)

Clearing the angle we have:

x = Atan (4/6)

x = 19.77

Rounding off we have:

x = 19.70 degrees

What is the slope of the line

Answers

Answer:

4/5

Step-by-step explanation:

There are so many points on the graph whose coordinates are impossible to estimate. Look for two "nice" points--points with integer coordinates. The line passes through (-2, 0) and (3, 4).

Slope = rise / run. It's the ratio of vertical change to horizontal change as you move from one point on the line to another.

Moving from (-2, 0) to (3, 4), the vertical change is 4 - 0 = 4.

The horizontal change is 3 - (-2) = 5.

The slope is 4/5

What is the next term of the arithmetic sequence?
-2,3, 8, 13, 18,

Answers

Answer:

23

Step-by-step explanation:

difference between number is 5

5n-7 to get to next term

What’s the correct answer for this?

Answers

I’m pretty sure it’s B.

Ammariah has finished 17/20 of her homework. How much more does she have to complete.
A.15/100
B.6/10
C.3/10
D.6/20
EXPLAIN

Answers

Answer: A 15/100

Step-by-step explanation:

1. Convert 17/20 into percents

2. You get 85% Which is how much of her homework she has completed

3. To find what she has left subtract 85 from 100

4. You get 15 or 15/100 or 15% more of the test she hasn’t completed and needs to complete

Please help ASAP! Will give BRAINLIEST! Please read the question THEN answer correctly! No guessing.

Answers

Answer:

A. X=6 only

Step-by-step explanation:

1 min to fill 16.50 gallons what is the constant

Answers

Answer:constant is 2/33

Step-by-step explanation:

1 min to fill 16.50 gallons

Time(t) varies directly as volume(v)

t α v

t =k x v Where k is constant

1=k x 16.50

k=1/16.50

k=2/33

Constant is 2/33

Simplify the expression below.
14a8y3 - 7 Ay5 + 28a12y2
7aty
A.
OB.
2a²y3 - ay5 + 4a3y2
2a4y? - JA + 428 y
2a4y3 – 5 + 428 y?
D. 2012,4 - 2876 +4215,3
C.

Answers

Answer:

14a8y3 - 7 Ay5 + 28a12y2-  7ay2 • (4a11 + 2a7y - y3)

Step-by-step explanation:

Equation at the end of step  1  :

 (((14•(a8))•(y3))-(7a•(y5)))+((22•7a12)•y2)

Step  2  :

Equation at the end of step  2  :

 (((14•(a8))•(y3))-7ay5)+(22•7a12y2)

Step  3  :

Equation at the end of step  3  :

 (((2•7a8) • y3) -  7ay5) +  (22•7a12y2)

Pull out like factors

Answer:   7ay2 • (4a11 + 2a7y - y3)

Hope this helps.

Fie triunghiul ABC isoscel cu AB=AC=3 cm daca mediatoarea laturi AC intersectat cu latura BC in M si perimetrul thriunghiului AMC=12 cm.Calculati MC

Answers

Answer:

MC = 4.5cm

Step-by-step explanation:

Question:

Let the isosceles triangle ABC with AB = AC = 3 cm. if the mediator of the sides AC intersects with the side BC in M ​​and the perimeter of the triangle AMC = 12 cm. Calculate MC.

Solution:

Find attached the diagram used in solving the question.

Given:

∆ABC is an isosceles triangle (two sides and angles are equal)

AB = BC = 3cm

Perimeter of ∆AMC = 12cm

From the diagram, M cuts AC at the the middle.

AD = CD = AC/2 = 3/2

Perimeter of Right angled ∆AMD = AM + AD + MD

= 3/2 + AM +MD

Perimeter of Right angled ∆CMD =CM + CD + MD

= 3/2 + CM +MD

Right angled ∆AMD = Right angled ∆CMD

CM = AM

Therefore ∆AMC is an isosceles triangle

CM = AM (two sides of an isosceles triangle are equal)

Let CM = AM = x

Perimeter of ∆AMC = AM + CM + AC

12 = x + x + 3

12 = 2x + 3

2x = 12-3

2x = 9

x = 9/2 = 4.5

CM = AM = 4.5cm

MC = CM = 4.5cm

Need help on this assignment it is due VERY SOON

All have something to do with area of composite figures. Please help. I'd be very greatful.

Answers

Answer

the first photo answer is 174 (2nd photo is A) Answer to 3rd photo is 72. The last photo is 8100 ft²

Step-by-step explanation:

1st Photo

12 x 12 is 144

5 times 12 divided by 6 - (5 x 6) 30

144 plus 30 is 174

2nd Photo

3 x 5 15

15 is A

3rd Photo

6 x 12 is 72

72 for 3rd Photo

4th Photo

90 x 90 is

8,100

A cube with side length mmm has a volume of 343343343 cubic centimeters. The following equation shows the volume of the cube.
m^3 = 343m
3
=343m, cubed, equals, 343
What is the side length of the cube in centimeters?

Answers

Answer:7cm

Step-by-step explanation:

m^3=343

Take the cube root of both sides

m=7

Needing help can’t seem to find the answers? ‍♀️

Answers

Step-by-step explanation:

19. Kim's Path = √240² + 100²

= √57600 + 10000

= √67600

= 260 ft.

Answer:

17 question: we are going to use Pythagorean Theorem :

33^2 + 56^2 = x^2

1089+3165 = x^2

4225=x^2

x=-65

x=65

the answer is c)65

18 question i dont know :((

19 question : we are going to use same method as the first question : the answer is 260 ft

that's all that i can do for you

Step-by-step explanation:

hope this helps you :)

Sam conducted a survey to determine your favorite flavor of ice cream is his hometown. 15 of the 50 people he surveyed liked strawberry best. How many of the 3,500 residents in the town would you expect to like strawberry best

Answers

Answer: 1050 people.

Step-by-step explanation: It's a rule of three question.

15/50 as x/3500

Then you have that 50x = 52500

x = 1050

There you have it.

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