Consider the following function.
f(x) = 1 + 5/x - 4/x2
(a) Find the vertical asymptote(s).
(b) Find the interval where the function is increasing.
Find the interval where the function is decreasing.
(c) Find the local maximum and minimum values.
(d) Find the interval where the function is concave up.
Find the interval where the function is concave down.
Find the inflection point.
(e) Use the information from parts (a)-(d) to sketch the graph of f.
Answers
Answer:
See below
Step-by-step explanation:
I assume the function is [tex]f(x)=1+\frac{5}{x}-\frac{4}{x^2}[/tex]
A) The vertical asymptotes are located where the denominator is equal to 0. Therefore, [tex]x=0[/tex] is the only vertical asymptote.
B) Set the first derivative equal to 0 and solve:
[tex]f(x)=1+\frac{5}{x}-\frac{4}{x^2}[/tex]
[tex]f'(x)=-\frac{5}{x^2}+\frac{8}{x^3}[/tex]
[tex]0=-\frac{5}{x^2}+\frac{8}{x^3}[/tex]
[tex]0=-5x+8[/tex]
[tex]5x=8[/tex]
[tex]x=\frac{8}{5}[/tex]
Now we test where the function is increasing and decreasing on each side. I will use 2 and 1 to test this:
[tex]f'(2)=-\frac{5}{2^2}+\frac{8}{2^3}=-\frac{5}{4}+\frac{8}{8}=-\frac{5}{4}+1=-\frac{1}{4}[/tex]
[tex]f'(1)=-\frac{5}{1^2}+\frac{8}{1^3}=-\frac{5}{1}+\frac{8}{1}=-5+8=3[/tex]
Therefore, the function increases on the interval [tex](0,\frac{8}{5})[/tex] and decreases on the interval [tex](-\infty,0),(\frac{8}{5},\infty)[/tex].
C) Since we determined that the slope is 0 when [tex]x=\frac{8}{5}[/tex] from the first derivative, plugging it into the original function tells us where the extrema are. Therefore, [tex]f(\frac{8}{5})=1+\frac{5}{\frac{8}{5}}-\frac{4}{\frac{8}{5}^2 }=\frac{41}{16}[/tex], meaning there's an extreme at the point [tex](\frac{8}{5},\frac{41}{16})[/tex], but is it a maximum or minimum? To answer that, we will plug in [tex]x=\frac{8}{5}[/tex] into the second derivative which is [tex]f''(x)=\frac{10}{x^3}-\frac{24}{x^4}[/tex]. If [tex]f''(x)>0[/tex], then it's a minimum. If [tex]f''(x)<0[/tex], then it's a maximum. If [tex]f''(x)=0[/tex], the test fails. So, [tex]f''(\frac{8}{5})=\frac{10}{\frac{8}{5}^3}-\frac{24}{\frac{8}{5}^4}=-\frac{625}{512}<0[/tex], which means [tex](\frac{8}{5},\frac{41}{16})[/tex] is a local maximum.
D) Now set the second derivative equal to 0 and solve:
[tex]f''(x)=\frac{10}{x^3}-\frac{24}{x^4}[/tex]
[tex]0=\frac{10}{x^3}-\frac{24}{x^4}[/tex]
[tex]0=10x-24[/tex]
[tex]-10x=-24[/tex]
[tex]x=\frac{24}{10}[/tex]
[tex]x=\frac{12}{5}[/tex]
We then test where [tex]f''(x)[/tex] is negative or positive by plugging in test values. I will use -1 and 3 to test this:
[tex]f''(-1)=\frac{10}{(-1)^3}-\frac{24}{(-1)^4}=-34<0[/tex], so the function is concave down on the interval [tex](-\infty,0)\cup(0,\frac{12}{5})[/tex]
[tex]f''(3)=\frac{10}{3^3}-\frac{24}{3^4}=\frac{2}{27}>0[/tex], so the function is concave up on the interval [tex](\frac{12}{5},\infty)[/tex]
The inflection point is where concavity changes, which can be determined by plugging in [tex]x=\frac{12}{5}[/tex] into the original function, which would be [tex]f(\frac{12}{5})=1+\frac{5}{\frac{12}{5}}+\frac{4}{\frac{12}{5}^2 }=\frac{43}{18}[/tex], or [tex](\frac{12}{5},\frac{43}{18})[/tex].
E) See attached graph
Related Questions
A box contains 7 green marbles, 10 blue marbles, and 5 red marbles. Three marbles are selected at random from the box, one at a time, without replacement. Find the probability that two of the marbles selected are green and the other marble is red.
Answers
Answer:
5/253
Step-by-step explanation:
The first green is 7/23, the second is 6/22 and the red is 5/21. Multiply them together for 5/253.
What is the next whole number after 267 In base 8
Answers
The next number would be 270₈. The given number has an digit expansion of
267₈ = 2•8² + 6•8¹ + 7•8⁰
If x is the next number, then adding 1 to the both sides gives
x = 2•8² + 6•8¹ + 7•8⁰ + 1
x = 2•8² + 6•8¹ + 7•8⁰ + 1•8⁰
x = 2•8² + 6•8¹ + 8•8⁰
x = 2•8² + 6•8¹ + 1•8¹
x = 2•8² + 7•8¹
x = 2•8² + 7•8¹ + 0•8⁰
and so x = 270₈.
The square below represents one whole.
What percent is represented by the shaded area?
Answers
——————————————-
I need help with this question
Answers
Answer:
check online for more information
Mary had 60$ and spent 3/4 on a coat. plus she brought a skirt with 1/2 she had left . how much do she have left?
Answers
60 - 45 = $15 left
1/2 of $15 = $7.50 for the skirt
15 - 7.50 = $7.50 left
The following ordered pairs represent a function.
(-2,-4) (-1, -2) (0, 0) (1, 2) (2, 4)
Which two ordered pairs can be added to the given set and still have the set represent
a function?
Answers
Answer:
(-3,-6) and (3,6)
Step-by-step explanation:
First, use process of elimination on the answer choices with x coordinates already used in the relation, because a value of x cannot have 2 outputs, or y-values. This leaves you with 2 options. Hope I helped :D
A pianist charges $200 to perform at a restaurant. At the end of the night she collected her tips and left with a total earning of $310. Write an equation and solve for the amount she made in tips for the performance.
Answers
Answer:
310-200= T
Step-by-step explanation:
310 = 200 + x
Let’s break this down. 310 is their total earning, 200 is our set price for performing, and x represents how much she earned in tips.
310 = 200 + x
110 = x
Hope this helps!
4/k =8/3 what is k?
Answers
Answer:
1.5
Step-by-step explanation:
4 x 2 = 8, so k x 2 = 3
3 / 2 = 1.5
Okie help please math math math
Answers
Answer:
x = 15
Step-by-step explanation:
The angles shown are supplementary angles. Supplementary angles add up to equal 180°
This means that 2x + 8 + 8x + 22 must equal 180
Note that we've just created an equation that we can use to solve for x
We now solve for x using that equation
2x + 8 + 8x + 22 = 180
combine like terms : 2x + 8x = 10x and 8 + 22 = 30
10x + 30 = 180
subtract 30 from both sides : 30 - 30 cancels out and 180 - 30 = 150
10x = 150
divide both sides by 10: 10x / 10 = x and 150 / 10 = 15
we're left with x = 15
Answer:
x = 15
Step-by-step explanation:
The angles shown are supplementary angles. Supplementary angles add up to equal 180°
This means that 2x + 8 + 8x + 22 must equal 180
Note that we've just created an equation that we can use to solve for x
We now solve for x using that equation
2x + 8 + 8x + 22 = 180
combine like terms : 2x + 8x = 10x and 8 + 22 = 30
10x + 30 = 180
subtract 30 from both sides : 30 - 30 cancels out and 180 - 30 = 150
10x = 150
divide both sides by 10: 10x / 10 = x and 150 / 10 = 15
we're left with x = 15
Question 10 of 10
To model some data, Pat ran an exponential regression and found the
regression equation to be y = 0.496.1.3%,
The scatterplot and residual plot are shown.
Scatterplot
Residual plot
y
4
2
Select the true statements.
O A. The model is a good fit because the patterns in the plots are both
curves.
B. The points in the scatterplot follow a line, so a linear model would
fit the data better.
O C. The residual plot changes direction, so a quadratic model would
fit the data better.
D. The model is not a good fit because there is a pattern in the
residual plot.
Answers
Answer:
B.)The points in the scatterplot follow a line, so a linear model would fit the data better
A.)The model is not a good fit because there is a pattern in the residual plot.
Step-by-step explanation:
The linear model would best fit the scatter plot while the quadratic model would fit the residual plot. The true statements are: B and C.
What is a Linear Model and a Quadratic Model?A linear model is appropriate for data points that lie along a straight line on graph. If the line is not straight but curved and changes direction, a quadratic model is a better model for such data.
Thus, since the data points on the scatter plot are on a straight line, a linear model would fit the data.
For the residual plot that changes direction, a quadratic model would better fit the data.
The true statements are therefore: B and C.
Learn more about linear and quadratic model on:
https://brainly.com/question/13496810
#SPJ2
Martha has 2 apples if she buys 2 more how much does she have?
Answers
I Hope This Helps
help pls and thank you
Answers
Answer:
A. $10
Step-by-step explanation:
They make $10 profit an hour.
Complete the process of solving the equation.
Fill in the missing term on each line. Simplify any fractions.
9W + 1 = 19
9w =
Subtract 1 from both sides
W =
Divide both sides by 9
Answers
Answer:
see explanation
Step-by-step explanation:
9W + 1 = 19 ( subtract 1 from both sides )
9W = 18 ( divide both sides by 9 )
W = 2
n is m% of what?
It DEFINITELY a variable
Answers
Function 1 is defined by the equation y=-1/4r-5
Answers
Step-by-step explanation:
and what is function 2 ?
but I think I can risk a guess based on the little piece of the graph I can see.
the y-intercept is simply the point on the y-axis (that means x=0), where the line crosses.
in other words, it is the y-value when x = 0.
the y-intercept of function 1 is therefore -5.
and the picture suggests that the visible line would cross the y-axis at about y=-6.
as -5 is greater than -6, function 1 has the greater y-intercept.
PLEASE HELP RELATIONSHIPS IN TRIANGLES
Answers
Answer
TN=22
Step-by-step explanation:
You know that MT+TP+TN using the Orthocenter thereom.
For lunch, I ate a toasted cheese sandwich and it has 450 calories and 16g protein with some baked potato fries and it has 230 calories and 3g protein. How much protein and calories did I eat in all?
Answers
Answer:
Give the person above me brainliest
Step-by-step explanation:
graph the relation y=2x
Answers
An airplane can ascend at a rate of 50 1/2 meters in 2/3 of a second. How many meters can the airplane ascend in one second?
Answers
Answer:
78.75
Step-by-step explanation:
Evaluate the function when x = -1,0, and 4.
9. g(x) = 3 -
X
g(-1) =
g(0) =
+
g(4) =
Plz help me I need the answer
Answers
G(0) = 3
G(4) = 1
If y’all show y’all work and do all the questions I add extra extra points
Answers
Answer:
For 5.) I believe the slope is 3/5, for 6.) the slope is -2/8 (-1/4 when simplified) and the y-int is 2, for 7.) the slope is 6/2 (3/1 when simplified), for 8.) the slope is -2/4 (-1/2 when simplified) + the y-int is -3, and for 9.) the slope is 1/6.
Step-by-step explanation:
5.) the slope is 3/5 because it rises 3 and runs 5 (from the two plotted points)
6.) the slope is -2/8 (-1/4) because from the two points, it runs 8 and drops 2 (so -2) + when simplified its -1/4 because -2/8 divided by 2 on top and bottom would get you -1/4. And y-int is 2 because on the y-axis, the line is going through 2.
7.) the slope is 6/2 (3/1) because it rises 6 points and runs 2 (from the two points) and when simplified (6/2 divided by 2 on both top and bottom) would get you 3/1.
8.) the slope is -2/4 (or -1/2) because it runs 4 and drops 2 (which would be -2), and when simplified it gets you -1/2.
9.) the slope is 1/6 because between the two points, you rise one and run 6. Which gets you 1/6
Not sure if these are all correct, but I hope this helps :)!
A courtyard has the shape of a rectangle 45 ft and 28 ft wide. How long is a sidewalk from one corner diagonally to the opposite corner?
56 ft
47 ft
50 ft
53 ft
Answers
Answer:
53
Step-by-step explanation:
We use the pythagorean theorem, a^2 + b^2 = c^2. We can fill a and b for 45 and 28. 45^2 is 2025, and 28^2 is 784. Adding them up gives us 2809. The square root of 2809 is 53.
What is 12,925 to the nearest ten thousand
Answers
Answer:
To round to the nearest 10,000, you look at the number immediately to the right. If this number is 5 or above, you add 1 to the column in question and zero all numbers to the right. If the number immediately to the right is 4 or less, you just zero all numbers to the right.
In this case, the 10,000 column is the column with the 1. The number immediately to the right is 2. This is 4 or less, so we set all numbers to the right of the 10,000 column to zero.
I get 10,000
Step-by-step explanation:
Hope this helps!
A deposit of $1000 is made in an account that earns 8% interest compounded quarterly. The balance in the account after n quarters is given by the following sequence. Find the balance in the account after ten years.
a
=100.000)
n=1,2,3
The balance after ten years.
Answers
The balance after ten years is $2,208.04.
When an amount of money is compounded quarterly, it means that the amount deposited and the interest earns interest every quarter. The formula that can be used to determine the balance in 10 years is:
FV = P (1 + r)^nm
FV = Future value P = Present value = $1000 R = interest rate = 8%/4 = 2% m = number of compounding = 4 N = number of years = 10$1000 x (1.02)^40 = $2,208.04
To learn more about future value, please check: https://brainly.com/question/18760477
Integrate:Cos 7x/(1+sin**2(7x))
Answers
Answer:
7x/1+7sin(2)x
Step-by-step explanation:
A force of 18 lb is required to hold a spring stretched 2 in. beyond its natural length. How much work W W is done in stretching it from its natural length to 7 in. beyond its natural length
Answers
Answer:
Step-by-step explanation:
Average force is (19 + 0)/2 = 9 lb
2 inches = 1/6 ft
W = Fd = 9(1/6) = 1.5 ft•lb
How do u write x/7=14 in algebra tiles
Answers
Answer:
98 is the answer.
Step-by-step explanation:
x/7 = 14
=> 14 × 7 = 7 × x/7
=> x = 98
Therefore, 98 is the answer.
Hoped this helped.
There are 4 dogs allowed at the park when there are 6 people. How many dogs are
there when there are 9 people?
Answers
Answer:
7
Step-by-step explanation:
since there are 4 dogs for every 6 people,
if we add 3 people, there will be 9 people and 7 dogs.
Answers plz fast ????? :D
Answers
Ans
Step-by-step explanation:
Help use photo for questions
Answers
y=-6/3
