Let F(x)=∫
t−3
t
2+7
for − ∞ < x < ∞
x
0
(a) Find the value of x where F attains its minimum value.
(b) Find intervals over which F is only increasing or only decreasing.
(c) Find open intervals over which F is only concave up or only concave down.

Answers

Answer 1

(a) It looks like you're saying

[tex]\displaystyle F(x) = \int_0^x (t - 3t^2 + 7) \, dt[/tex]

Find the critical points of F(x). By the fundamental theorem of calculus,

F'(x) = x - 3x² + 7

The critical points are where the derivative vanishes. Using the quadratic formula,

x - 3x² + 7 = 0   ⇒   x = (1 ± √85)/6

Compute the second derivative of F :

F''(x) = 1 - 6x

Check the sign of the second derivative at each critical point.

• x = (1 + √85)/6 ≈ 1.703   ⇒   F''(x) < 0

• x = (1 - √85)/6 ≈ -1.370   ⇒   F''(x) > 0

This tells us F attains a minimum of

[tex]F\left(\dfrac{1-\sqrt{85}}6\right) \approx \boxed{-6.080}[/tex]

(b) Split up the domain of F at the critical points, and check the sign of F'(x) over each subinterval.

• over (-∞, -1.370), consider x = -2; then F'(x) = -7 < 0

• over (-1.370, 1.703), consider x = 0; then F'(x) = 7 > 0

• over (1.703, ∞), consider x = 2; then F'(x) = -3 < 0

This tells us that

• F(x) is increasing over ((1 - √85)/6, (1 + √85)/6)

• F(x) is decreasing over (-∞, (1 - √85)/6) and ((1 + √85)/6, ∞)

(c) Solve F''(x) = 0 to find the possible inflection points of F(x) :

F''(x) = 1 - 6x = 0   ⇒   6x = 1   ⇒   x = 1/6

Split up the domain at the inflection point and check the sign of F''(x) over each subinterval.

• over (-∞, 1/6), consider x = 0; then F''(x) = 1 > 0

• over (1/6, ∞), consider x = 2; then F''(x) = -11 < 0

This tells us that

• F(x) is concave up over (-∞, 1/6)

• F(x) is concave down over (1/6, ∞)


Related Questions

hlep me it is easy plssssssssssssss

Answers

Answer:

C. (10 - 2) ÷ 4

Step-by-step explanation:

To determine which expression represent a number that is one fourth as great as 10-2 the following logical reasoning has to be made:

Given that a number is raised that is a quarter as large as (10-2), the result of this mathematical operation must be divided by 4.

Thus, since 10 minus 2 is equal to 8 and 8 divided by 4 gives 2, the division is necessary as a mathematical operation.

So, since the only option that raises a division is option C, that's the correct option.

Find the following:

sin A
cos A
tan A
cot A
sec A
csc A

Answers

Step-by-step explanation:

we use the law of sine :

sin(A)/a = sin(B)/b = sin(C)/c

with the related angles being opposite of the lines.

so, we have

sin(A)/1 = sin(A) = sin(90)/ sqrt(10) = 1/ sqrt(10) =

= 0.316227766...

cos²(A) + sin²(A) = 1

cos(A) = sqrt(1 - sin²(A)) = sqrt(1 - 0.1)) = sqrt(0.9) =

= 0.948683298...

tan(A) = sin(A)/cos(A) = 0.333333333...

cot(A) = 1/tan(A) = 3

sec(A) = 1/cos(A) = 1.054092553...

csc(A) = 1/sin(A) = 3.16227766...

can someone plzzz help me with this (multiple choice ) i have more if anyone is good at this

Answers

A: Hope it helps Because…………………… I don’t really know I know it’s a I did this

Help help again ..2 more :/ thank you again ..

Answers

8^2 + 12 - 2(8)
64 + 12 - 16
76 - 16
60
so your answer would be 60

the area of a rectangle and square are same . if area of the square is 900 cm² and length of the rectangle is 45cm find the perimeter of the rectangle.​

Answers

Answer: 130cm

Let's think the width of the rectangle as, "x"

45*x= 900

x= 900/45

x= 20

45+45+20+20

= 130cm

3. Which of the following is a rational
number that is not an integer?
A-2
C 0
B -0.5
D 4

Answers

Step-by-step explanation:

-0.5 cause decimals and fraction are not included

According to his running log, Baldwin averaged 4 miles per week last month and 25% more this month. How much did he average this month?

Answers

4(1.25) = if it’s only one month then it’s 5 miles

Answer:

4(1.25) = if it’s only one month then it’s 5 miles

Step-by-step explanation:

Please mark brainliest

help plsssssssssssssss

Answers

Answer:step 2

Step-by-step explanation:

SHOW YOUR WORK!!! PLEASE HELP!!! ASAP!!!!

Answers

Answer:

84

Step-by-step explanation:

TPS + SPR = RPT and TPS = RPT - SPR

The measure of RPT and SPR are given so we just need to place them and do the calculation

TPS = 137 - 53

TPS = 84

another fraction problem. Have tried this one a couple of times and cant find fault in what i did

Answers

Answer:

Perimeter = 12 1/2 in

Area = 9.375 in

Step-by-step explanation:

Perimeter:

(3 3/4) + (3 3/4) + (2 1/2) + (2 1/2) = 12 1/2

Area:

(3 3/4) * (2 1/2) = 9.375

Answer:

P = 12 1/2 in

A = 9 3/8 in²

Step-by-step explanation:

L = 3 3/4 in

W = 2 1/2 in

Area:

A = L × W

A = (3 3/4 in)(2 1/2 in)

A = 15/4 in × 5/2 in

A = 75/8 in²

A = 9 3/8 in²

Perimeter:

P = 2(L + W)

P = 2(3 3/4 in + 2 1/2 in)

P = 2(15/4 in + 5/2 in)

P = 2(15/4 in + 10/4 in)

P = 2(25/4 in)

P = 25/2 in

P = 12 1/2 in

anyone who solves this problem for me .... I really need help

Answers

a) The value of [tex]k'(0)[/tex] is [tex]\frac{3\sqrt{3}}{2}[/tex].

b) The value of [tex]m'(5)[/tex] is approximately -0.034.

c) The value of [tex]x[/tex] is approximately 0.622.

a) [tex]f(x)[/tex] is a piecewise function formed by two linear functions, whose form is defined by the following definition:

[tex]f(x) = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \cdot x + b[/tex] (1)

Where:

[tex](x_{1}, y_{1})[/tex], [tex](x_{2}, y_{2})[/tex] - Two distinct points of the line.[tex]x[/tex] - Independent variable.[tex]f(x)[/tex] - Dependent variable.[tex]b[/tex] - x-Intercept

Now we proceed to determine the linear functions:

Line 1: [tex]x \in [-1, 2)[/tex]

[tex](x_{1}, y_{1}) = (0, 3)[/tex], [tex](x_{2}, y_{2}) = (1, 5)[/tex], [tex]b = 3[/tex]

[tex]f(x) = \frac{5-3}{1-0}\cdot x + 3[/tex]

[tex]f(x) = 2\cdot x + 3[/tex]

Line 2: [tex]x \in [2, 8][/tex]

[tex](x_{1}, y_{1}) = (2, 7)[/tex], [tex](x_{2}, y_{2}) = (8, 3)[/tex]

First, we determine the slope of function:

[tex]m = \frac{3-7}{8-2}[/tex]

[tex]m = -\frac{2}{3}[/tex]

Now we proceed to determine the intercept of the linear function:

[tex]7 = -\frac{2}{3}\cdot 2 + b[/tex]

[tex]b = \frac{25}{3}[/tex]

[tex]f(x) = -\frac{2}{3}\cdot x +\frac{25}{3}[/tex]

The first derivative of a linear function is its slope, and the first derivative of a product of functions is defined by:

[tex]k'(x) = f'(x)\cdot g(x) + f(x)\cdot g'(x)[/tex]

If we know that [tex]f(x) = 2\cdot x + 3[/tex], [tex]f'(x) = 2[/tex], [tex]g(x) = \sqrt{x^{2}-x+3}[/tex], [tex]g'(x) = \frac{2\cdot x - 1}{2\cdot \sqrt{x^{2}-x+3}}[/tex] and [tex]x = 0[/tex], then:

[tex]k'(x) = 2\cdot \sqrt{x^{2}-x+3}+\frac{(2\cdot x +3)\cdot (2\cdot x - 1)}{2\cdot \sqrt{x^{2}-x+3}}[/tex]

[tex]k'(0) = 2\sqrt{3}-\frac{3}{2\sqrt{3}}[/tex]

[tex]k'(0) = \frac{12-3}{2\sqrt{3}}[/tex]

[tex]k'(0) = \frac{9}{2\sqrt{3}}[/tex]

[tex]k'(0) = \frac{3\sqrt{3}}{2}[/tex]

The value of [tex]k'(0)[/tex] is [tex]\frac{3\sqrt{3}}{2}[/tex].

b) The derivative is found by means of the formulas for the derivative of the product of a function and a constant and the derivative of a division between two functions:

[tex]m'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{2\cdot [g(x)]^{2}}[/tex] (2)

If we know that [tex]f(x) = -\frac{2}{3}\cdot x +\frac{25}{3}[/tex], [tex]f'(x) = -\frac{2}{3}[/tex], [tex]g(x) = \sqrt{x^{2}-x+3}[/tex], [tex]g'(x) = \frac{2\cdot x - 1}{2\cdot \sqrt{x^{2}-x+3}}[/tex] and [tex]x = 5[/tex], then:

[tex]f(5) = 5[/tex]

[tex]f'(5) = -\frac{2}{3}[/tex]

[tex]g(5) = \sqrt{23}[/tex]

[tex]g'(5) = \frac{9\sqrt{23}}{46}[/tex]

[tex]m'(5) = \frac{\left(-\frac{2}{3} \right)\cdot \sqrt{23}-\left(-\frac{2}{3} \right)\left(\frac{9\sqrt{23}}{46} \right)}{3\cdot 25}[/tex]

[tex]m'(5) \approx -0.034[/tex]

The value of [tex]m'(5)[/tex] is approximately -0.034.

c) In this case we must find a value of [tex]x[/tex], so that [tex]h'(x) = 2[/tex]. Hence, we have the following formula below:

[tex]5\cdot e^{x}-9\cdot \cos x = 2[/tex]

A quick approach is using a graphing tool a locate a point so that [tex]5\cdot e^{x}-9\cdot \cos x = 2[/tex]. According to this, the value of [tex]x[/tex] is approximately 0.622.

To learn more on derivatives, we kindly invite to check this verified question: https://brainly.com/question/21202620

Image attached giving 25 points please help

Answers

Answer:

i think x = -5, y = -7 is a other solution set

Which graph could potentially have a correlation coeffiecient (r value) of 0.95?

Answers

Answer:

C,

Step-by-step explanation:

A and D is incorrect because they are negative correlation

and B is incorrect because a correction of 0.95 means all values are close to each other, but B values are scattered much more that C values

so answer C


What is the formula for calculating momentum?

Answers

Answer:

 (c)  p = vm

Step-by-step explanation:

Momentum, represented by 'p', is the product of mass and velocity. It is a vector quantity, just as velocity is a vector quantity.

  p = vm

A waitress kept track of whether her customers ordered an appetizer and
dessert. Her data are shown in a relative frequency table.
Appetizer
Total
No appetizer
0.3
Dessert
0.1
0.4
No dessert
0.2
0.4
0.6
Total
0.3
0.7
1.0
What does the 0.1 in the highlighted cell mean?
A. 10% of her customers ordered dessert.
B. 10% of the customers who ordered an appetizer ordered dessert.
C. 10% of her customers ordered an appetizer and dessert.
D. 10% of her customers ordered an appetizer.

Answers

According to the two-way table, the correct option is:

10% of her customers ordered an appetizer and dessert.

From the table, it is taken that:

0.1 = 10% of the customers ordered both an appetizer and a dessert.0.3 = 30% of the customers ordered a dessert but not an appetizer.0.4 = 40% of the customers ordered a dessert.0.2 = 20% of the customers ordered an appetizer but not a dessert.0.4 = 40% of the customers ordered neither a dessert nor an appetizer.0.6 = 60% of the customers did not order a dessert.0.3 = 30% of the customers ordered an appetizer.0.7 = 70% of the customers did not order an appetizer.The total is of 100%.

Hence, the highlighted 0.1 = 10% means that:

10% of her customers ordered an appetizer and dessert.

You can learn more about two-way tables at https://brainly.com/question/24670062

Luigi's Fine Dining charges $32.09 for its prix fixe meal plus $39.26 for every drink, d, you purchase. The equation that models the total cost (T) is T(d )= 39.26d + 32.09 Which value in the equation represents the rate of change?

A .. The number of drinks purchased
B .. $32.09
C .. $39.26
D .. $71.35

Answers

Answer:

C. $32.09, since it's not the fixed cost, it changes based on how much drinks are purchased.

A
It depends on the amount of drinks purchased to change the bill

what word phrase can you use to represent the algebraic expression 6x+6

Answers

Answer:

the word phrase use to represent 6x+ 6 is sum 6x and 6

Step-by-step explanation:

first you sum 6x and 6

1) 0.52 + 1.6 + 8.26 =
2) 1.4 + 5.98 + 9 + 0.39 =
3) 4.45 + 7 + 0.049 =
4) 12.54 – 1. 054 =
5) 0. 685 – 0. 5903 =
6) (34. 89) (0.875) =
7) (840) (0.625) =
8) 28.5 ÷ 0.87 =
9) 104 ÷ 6.4 =

Answers

Step-by-step explanation:

1 10.38

2 16.77

3 11.499

4 11.486

5 0.0947

6 30.52875

7 525

8 32.57862069

9 16.5625

6 lb = how many oz .​

Answers

Answer:

96 oz

Step-by-step explanation:

1 lb= 16 oz

so, 16×6=96

solve for y and x
2+3x-1=4y
And
2x-3=y

Answers

Answer:

x = 2.6, y = 2.2

Step-by-step explanation:

if 4y = 2 + 3x - 1 and

y = 2x - 3,

4y = 2 + 3x -1 and 4y = 8x - 12 so

8x - 12 = 2 + 3x - 1

so 5x - 12 = 1,

5x = 13,

x = 2.6

If x = 2.6,

y = 5.6 - 3

so y = 2.2

Please mark me as Brainliest!

A cafeteria sells 30 drinks every 15 minutes. How many drinks can be sold in one hour?

Answers

Answer:30 drinks

Step-by-step explanation:

Choose the symbol that correctly compares the fractions below.

Answers

Answer:

[tex] \frac{2}{5} > \frac{1}{6} [/tex]

Answer: B. >

#CarryOnLearning

1plss help, its time

Answers

Answer:

60 degrees

Step-by-step explanation:

Similar shapes have the same angles but different side lengths.

Answer:

The answer is 60°

Step-by-step explanation:

Because pentagon JKLMN is basically the same as pentagon VWXYZ and if you see in the smaller pentagon the angle for L is 90° and same goes for the bigger pentagon

Use the GCF and the Distributive property to find the sum of 66+78

Answers

Here is ur answer of your question

100 points will award brainliest crown
Draw the image of EFG under the dilation with scale factor 3/2 and center of dilation (0, 0). Label the image E' F' G'

Answers

Answer:

Should be 3 units away from the picture

Step-by-step explanation:

I hope this helps! ;)

The coordinates of the image of EFG under the dilation with scale factor 3/2 and center of dilation (0, 0) are E(-3,3), F'(-3,-3) and G'(3,-3).

What is the dilation of the figure?

Dilation of a figure, means the transformation of the figure. The factor by which the given figure dilated, called the scale factor of dilation.

The image of EFG under the dilation with scale factor and center of dilation has to be drawn.

The value of scale factor of dilation is 3/2.The center of dilation is at (0, 0).

The coordinates or the triangle are E(-2,2), F(-2,-2) and G(2,-2). The new points after the dilation are,

[tex]E'(-2\times\dfrac{3}{2},2\times\dfrac{3}{2})=E'(-3,3)\\F'(-2\times\dfrac{3}{2},-2\times\dfrac{3}{2})=F'(-3,-3)\\G'(2\times\dfrac{3}{2},-2\times\dfrac{3}{2})=G'(3,-3)[/tex]

The dilated image of triangle EFG is attached below.

Thus, the coordinates of the image of EFG under the dilation with scale factor 3/2 and center of dilation (0, 0) are E(-3,3), F'(-3,-3) and G'(3,-3).

Learn more about the dilation of the figure here;

https://brainly.com/question/3457976

#SPJ2

solve using system of equations by using elimination method for -5x+4y=2 ; 9x-4y=6

Answers

Answer:

y = 3

x = 2

Step-by-step explanation:

-5x + 4y = 2

9x - 4y = 6

Sum both eq.

-5x + 9x = 4x

+4y - 4y = 0

2 + 6 = 8

then:

4x + 0 = 8

4x = 8

x = 8/4

x = 2

from the first eq.

-5x + 4y = 2

-5*2 + 4y = 2

-10 + 4y = 2

4y = 2 + 10

4y = 12

y = 12/4

y = 3

Check:

from the second eq.

9x - 4y = 6

9*2 - 4*3 = 6

18 - 12 = 6

The students at Vienna Elementary sold donuts every day at school for 6 months. The table below shows the earnings for the first 6 weeks. If the pattern continues, how much will the students make in week 8?

Pics here

Answers

$100 because it goes 110,100,90,110,100,90

The student made $100 in week 8.

What is an expression?

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Amount earned from the first week to 6 weeks.

110, 100, 90, 110, 100, 90

We see that,

Every 3 weeks the amount gets repeated.

So,

7th week = 110

8th week = 100

Thus,

The amount earned in week 8 is $100.

Learn more about expressions here:

https://brainly.com/question/3118662

#SPJ2

LAST ATTEMPT MARKING AS BRAINLIEST!! ( write a rule to describe each transformation)

Answers

Answer:

See explanation

Step-by-step explanation:

Dilation of 4

G (0, 1) G' (0, 4)

F (1, 1) F' (4, 4)

Gx'/Gx = 0 / 0 = 0

Gy' / Gy = 4 / 1 = 4

Fx' / Fx = 4 / 1 = 4

Fy' / Fy = 4 / 1 = 4

What is 0.459 roused to the nearest hundredth

Answers

Answer:

0.459

the 9 rounds the 5 up to a 6 so its 0.46

Answer:

The answer is 0.46

Step-by-step explanation:

The 9 is over 5, so you round up which bumps the 5 up to a 6. It could also be written as 0.460 but the extra 0 is unnecessary.

Directions: Compare the sides and angles by filling in the blank with a < or > symbol.

Answers

Answer:

Number 21 is < number 23 is > number 22 is < and number 24 is >

Step-by-step explanation:

Other Questions
Given 7 is one of the roots of the quadratic equation(x + k)2 = 16, where k is a constant, find the values of k. Joe and Zoya are lawyers. They have agreed upon starting a law firm, where they intend to share the profits and losses in the ratio of 3:2. What is this business ownership known as?a. partnershipb. sole proprietorshipc. corporationd. limited company what is10 + 9 i really dont know can someone help me suppose you want to change your school and join another one which is very expensive.make a list of the possible risks while joining another school. The table shows a function. Is the function linear or nonlinear? Do family psychologists help couples? which of the following words does not fill in the blank correctly? mis primos ____________ informacin. se conectan buscan imprimen guardan The above political cartoon indicates that Hitler was able to come topower because...2 pointsO the Nazi party helped Germany get a fair deal after WWI with the Treaty of Versailles.Othe Nazi party was the one political party that supported no longer abiding by theTreaty of Versaillesthe Nazi party was the one political party that agreed to still abide by the Treaty ofVersaillesO the Nazi party helped Germany win WWI. WILL GIVE BRAINLIESTy = 2x - 9y = 1/2x + 3These two lines are:parallelperpendicularneither someone please answer this How does critical thinking relates to peer assessment? what is amoxicillin and clavulanate potassium used for Find the measure of angle m PLEASE HELP SOMEBODY!!!!!!!!!! I uploaded a picture below. Please help!!No linksPicture is included What is the main difference between migration, immigration, and emigration? PLEASEEEEE HELPPPP 50 POINTS!! I need help on math true and false please Please help me do this with my exam tomorrow.Indicate the target item of the time to use, preserve and turn it in time. In your family and locality, what methods of harvesting, management and processing do you apply? Ex example Reread this excerpt from Sidney's Astrophil andStella, Sonnet 19," keeping in mind the theme: "peoplein love will continue loving, even if it makes themseem foolish."Select the two groups of lines that show the turn, orvolta, in the sonnet.