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
Would 5/5x2 3/4 be less greater or equal to 2 3/4
Answers
Answer:
greater
Step-by-step explanation:
if you add them and do the multiplication it is greater
Answer:
It would be equal
Step-by-step explanation:
You first divide 5/5, which is equal to 1, multiply that by 2 to get 2.
After that, there is a 3/4 sitting around so you add 3/4 to 2.
Then you will see that both are equal.
PLEASE ANSWER ILL GIVE 5 STARS AND A HEART
Answers
Answer:
25
Step-by-step explanation:
10 + (b - a) · 5
10 + (7 - 4) · 5
10 + 3 · 5
10 + 15 = 25
Answer:
25
Step-by-step explanation:
If a = 4 and b = 7
First you gotta plug it in, so it'd become 10+(7-4)×5
Next we gotta use PEMDAS. So it would be 7-4 which would be 3, multiply that by 5 which would make 15, then add 10 for 25.
What is the missing constant term in the perfect square that starts with x^2+2x?
Answers
Answer:
+ 1
Step-by-step explanation:
Given
x² + 2x
Using the method of completing the square
add ( half the coefficient of the x- term)² to x² + 2x
= x² + 2(1)x + 1²
= x² + 2x + 1
= (x + 1)² ← perfect square
c. What is located at the reflection of (3,1) across
the y-axis? What are the coordinates of this
location?
Answers
Check the picture below.
what is the exact value of tan(-π/3)?
A. -√3
B. -√3/3
C.√3/3
D. √3
Answers
[tex]\tan \left(-\dfrac{\pi}3\right)\\\\=-\tan \left(\dfrac{\pi}3\right)\\\\=-\sqrt 3[/tex]
Charles wants to buy an anime book that is on sale for $14.99. Her sister kyra wants to buy a novel book for $17.99. Both of them has to pay a 7% tax. Kyra has a coupon for 15% discount. Whats the difference between the amount that each siblings has to pay?
Answers
Answer:
The difference between their total amounts is .34 cents
also it didnt look like i was supposed to count a discount for charles so i didnt :)
Step-by-step explanation:
Charles14.99 times .07 = 1.04
14.99 + 1.04 = 16.03
Kyra17.99 times .15 = 2.69
17.99 -2.69 = 15.30
15.30 times .7 = 1.07
15.30 + 1.07 = 16.37
16.37 - 16.03 = .34
13. The function y = 40(x - 1) + 70 can be used to determine the cost in dollars
for renting a trampoline park for a birthday party for x hours. What is the rate of
change of the cost in dollars to rent the trampoline park with respect to time?
A
$30 per hour
$40 per hour
B
с
$60 per hour
D
$110 per hour
Answers
Step-by-step explanation:
the function actually is
y = 40(x - 1) + 70 = 40x - 40 + 70 = 40x + 30
still, the slope (the factor of x) is 40.
that means y (the cost) increases by 40 for every additional hour (x increases by 1).
so, the rate of change (slope) is 40/1 = $40 per hour.
What rational number represents a drop of 4 1/4 in?
Enter your answer in the box.
Answers
Answer:
The fraction 1/4 is a rational number.
It stands for the ratio between the integers 1 and 4.
Answer:
this is your answer
Step-by-step explanation:
this is your answer
Two containers designed to hold water are side by side, both in the shape of a cylinder. Container A has a diameter of 18 feet and a height of 15 feet. Container B has a diameter of 16 feet and a height of 18 feet. Container A is full of water and the water is pumped into Container B until Conainter B is completely full.
To the nearest tenth, what is the percent of Container A that is full after the pumping is complete?
Answers
Answer:
5.2%
Step-by-step explanation:
Volume of cylinder= πr²h
In Container A,
D= 18ft, r=18/2=9ft, h=15ft
volume= 22/7 × 9² × 15
= 26730/7 cubic ft.
In Container B,
D=16ft, r= 16/2=8ft, h=18ft
volume= 22/7 × 8² × 18
= 25344/7 cubic ft.
percent of amount of water left in Container A= percent of container A that is full after pumping
Amount of water left in Container A,
26730/7 - 25344/7 = 1386/7
Therefore, % of container A that is full= 1386/7 ÷ 26730/7 × 100
(1386×7×100)/(7×26730)= 5.185
To the nearest tenth= 5.2%
Find the slope of the line passing through the points (-7,8) and (-7,-6)
Find the slope of the line passing through the points (-9,-8) and (3,-8)
Answers
Answer:
1) undefined
2) 0
Step-by-step explanation:
Slope formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
Problem 1:
(x1, y1) = (-7, 8)
(x2, y2) = (-7, -6)
Plug those 2 points into the slope formula and solve:
[tex]m=\frac{-6-8}{-7--7}\\\\m=\frac{-14}{0}[/tex]
Here, you can't divide by 0, so this slope is undefined. An undefined slope would make a completely vertical line on a graph. In this case, the equation of the line would be just:
[tex]x=-7[/tex]
Problem 2:
(x1, y1) = (-9, -8)
(x2, y2) = (3, -8)
[tex]m=\frac{-8--8}{3--9}\\\\m=\frac{0}{12}\\\\m=0[/tex]
The slope is 0, and that would make a horizontal line on a graph. The equation would be:
[tex]y=-8[/tex]
You don't actually need to fully work out the slope that way. If you ever have 2 points with the same x-coordinate, the slope has to be undefined. If you ever have 2 points with the same y-coordinate, then the slope has to be 0, assuming it's linear of course. You can see that in the points given, the first 2 have the same x and the second 2 have the same y.
How many meters are equal to 18 kilometers?
Answers
Answer:
18000
Step-by-step explanation:
The number of meters in 18 kilometers is 18,000 meters(m)
How to convert meters to kilometers1900 meter(m) = 1 kilometer(km)x meters = 18 kilometers
x meters = 18 kilometers × 1000
x = 18,000 meters
Therefore, 18 kilometers is equivalent to 18,000 meters
learn more about conversion:
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Find the value of the expression
j − 14
for j = 18.
Answers
18 - 14 =
I think the answer is 4
Ali had 5/7 of a spool of yarn. He used 5/6 of his yarn for a project. What fraction of the spool was used for the project? Write your answer in simplest form.
- Due 11:59 today
- Will mark brainliest
Answers
Answer:
Step-by-step explanation:
We have no idea what time zone you are in so hope this gets there soon enough.
Do you mean "What fraction of a FULL spool of yarn did he use?" because he used 5/6 of the spool he had.
IF SO
Then
(5/7)(5/6) = 25/42
That's just under 60% of a full spool would be used to complete the same project again.
(x-3)^2=5 ? confused
Answers
[tex](x-3)^2=5\implies \sqrt{(x-3)^2}=\sqrt{5}\implies (x-3)=\sqrt{5}\implies x = \sqrt{5}+3[/tex]
A positive integer squared plus 3 times it’s consecutive integer is equal to 43
Answers
Answer:
the integer is 5
Step-by-step explanation:
x^2+3(x+1)=43
x^2+3x+3=43
x^2+3x+(9/4)=43-3+(9/4)
(x+(3/2))^2=42 1/4
x+(3/2)=sqrt(42 1/4)
x+(3/2)=6 1/2
x=(6 1/2)-(3/2)
x=5
check:
5^2+3(6)=43
25+18=43
43=43
Answer:
5
Step-by-step explanation:
let positive integer be x
x² + 3(x+1)=43
x²+3x+3-43=0
x² + 3x - 40 = 0
x² + 8x - 5x - 40 = 0
x(x+8) - 5(x+8)=0
(x-5)(x+8)=0
x-5=0 or x+8=0
x=5. or x=-8
but the question says positive integer,
therefore, x=5
Sammy and her family are on a trip. So far, they have traveled 225 miles and are 25% of the way to their destination. How many total miles will they travel?
Answers
Explanation: 25% is only 1/4 of the way, so in order to get to 100% you multiply it by 4. 225•4= 900
So 900 miles is your answer
A train has 11 train cars. At least 11 passengers are in each train car. No 3 neighboring train cars have more than 50 passengers altogether. What is the maximum possible number of people on the train?
Answers
The maximum possible number of people on the train is 183.
Given that a train has 11 train cars, and at least 11 passengers are in each train car, and no 3 neighboring train cars have more than 50 passengers altogether, to determine what is the maximum possible number of people on the train, the following calculation must be done:
50/3 = 16.66 16 + 17 + 17 = 50 16 + 17 + 17 + 16 + 17 + 17 + 16 + 17 + 17 + 16 + 17 = X 16 x 4 + 17 x 7 = X 64 + 119 = X 183 = X
Therefore, the maximum possible number of people on the train is 183.
Learn more about quantities in https://brainly.com/question/21477139
Answer:
189
Step-by-step explanation:
Set up 11 slots on a piece of paper, to simplify since each car must have 11 cars we will be alternating 28,11,11 in a repeated pattern thus getting us 28,11,11,28,11,11,28,11,11,28,11 when you add them up you get 189
At a school game, student tickets were 50 cents each and adult tickets were $1 each. If the total receipts from 900 tickets were $500, how many tickets of each kind were sold?
Answers
Answer:
Step-by-step explanation:
A(1.00) + S(0.50) = 500
A + S = 900
S = 900 - A
A(1.00) + (900 - A)(0.50) = 500
A + 450 - 0.5A = 500
0.5A = 50
A = 100 adults
S = 900 - 100
S = 800 students
No. of students tickets sold is 800 and no. of adult tickets sold is 100.
What is linear equation in two variable ?A linear equation has highest degree of 1 and it contains two variables generally x and y.
According to the given question At a school game, student tickets were 50 cents each and adult tickets were $1 each. If the total receipts from 900 tickets were $500.
Assuming no. of students to be x and no. of adults to be y.
∴ x + y = 900...(i)
And
0.5x + y = 500...(ii)
Subtracting eqn (i) and (ii) we get
0.5x = 400
x = 800.
So, 800 + y = 900
y = 100.
Therefore how many tickets are sold is same as the no. of students and adults which are 800 and 100 respectively.
learn more about linear equation here :
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A dog shelter is giving away 18 different dogs, but you have room for only 4 of them.
How many different dog families could you create?
Answers
Answer:
..2
Step-by-step explanation:
The number of different dog families that you can create will be 3060.
What are permutation and combination?A permutation is an act of correctly arranging things or pieces. Combinations are a method of taking stuff or pieces from an assortment of items or sets in which the order of the items is irrelevant.
The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.
A canine sanctuary is offering 18 unique canines, yet you have space for just 4 of them.
Then the number of different dog families that you can create is given as,
¹⁸C₄ = (18!) / [(18 - 4)! x 4!]
¹⁸C₄ = (18 x 17 x 16 x 15 x 14!) / (14! x 4 x 3 x 2 x 1)
¹⁸C₄ = 6 x 17 x 2 x 15
¹⁸C₄ = 3060
The number of different dog families that you can create will be 3060.
More about the permutation and the combination link is given below.
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Find the Simple interest paid for eight years on a $850 loan at 7.5% a year
Answers
Answer:
850 divided by 7.5
Step-by-step explanation:
Piper is located at (3, -12) on a map. Bobbi is located at
(3, 2). How far apart are they?
Answers
Check the picture below.
Can someone help me please I will mark u brilliant
Answers
Answer:
i believe it would be likely
Step-by-step explanation:
hope this helps! :D
have a miraculous day!! <3
(4x 2 − 2x + 10) + (−2x 2 + 4x − 2)
Answers
Answer:
2x²+2x+8
Step-by-step explanation:
4x²-2x+10-2x²+4x-2
=4x²-2x²-2x+4x+10-2
=2x²+2x+8
Question is in image below
A lamppost is 6 feet high and casts and 8 foot shadow. At the same time of day, a flagpole near the lamppost cast a 20 foot shadow. Using the properties of similar triangles, find the height, H, of the flagpole
Answers
Answer:
15 ft.
Step-by-step explanation:
I will assume that either both objects are standing straight. We can use similar triangles.
We can see that the side 8 is similar to the side 20, since they are both shadows. Therefore, we can set up an equation:
[tex]\frac{6}{8} =\frac{x}{20}[/tex]
we can cross multiply and get:
8x=120
solve for x:
x=15
Therefore, the lamppost must be 15 feet tall (unless it fell down but we won't talk about that)
When a retired police officer passes away, he leaves $325,000 to be divided among his five children and three grandchildren. The will specifies that each child is to get twice as much as each grandchild. How much does each get?
Answers
50.000
The high temperature was recorded as 25.6°F. The low temperature that same day was recorded as −2.7°F. What was the difference between the high and low temperatures for that day? Enter your answer as a decimal in the box.
Answers
28.3
Step-by-step explanation:
the difference is the answer to a subtraction problem so we can plug in our numbers like this
25.6-(-2.7) which equals 28.3
a good thing to remember when subtracting negative numbers is that a positive and negative won't always make a negative like in this example
Answer:
He's right it's 28.3
Step-by-step explanation:
Which number is a factor of 14 but not a multiple of 7??
Answers
Answer:
2
Step-by-step explanation:
Write out all the factors for 14
Then write out the multiples of 7
Compare the numbers then find the greatest number in 14's factors that isn't in 7's multiples
MARKING AS BRAINLIEST!! Starr if given functions are inverse)!
Answers
Answer: They are inverses of each other
==========================================================
Explanation:
We'll need to compute h(f(x)) and f(h(x)) to see if we get x each time.
[tex]h(x) = 9x-4\\\\h(f(x)) = 9(f(x))-4\\\\h(f(x)) = 9\left(\frac{x+4}{9}\right)-4\\\\h(f(x)) = x+4-4\\\\h(f(x)) = x\\\\[/tex]
and,
[tex]f(x) = \frac{x+4}{9}\\\\f(h(x)) = \frac{h(x)+4}{9}\\\\f(h(x)) = \frac{9x-4+4}{9}\\\\f(h(x)) = \frac{9x}{9}\\\\f(h(x)) = x\\\\[/tex]
Both composite functions lead to x each time.
The equations h(f(x)) = x and f(h(x)) = x being true means each function is the inverse of the other.
Fred bought a shirt that
was on sale. The original
price was $15.75, but it was
20% off. How much did
Fred save?
Answers
Answer:
Fred save about 3.15$ because is 20% off
If the Ending Inventory increases and everything else remains the same, what happens to the Food Cost?
Answers
An increase in the ending inventory will cause the cost of food sold at the end of the year to appear lower that it really is.
The cost of goods ( example, food) is directly affected by the amount of inventory at the end of an account period.
If the ending Inventory increases and everything else remains the same, the cost of goods sold is calculated as follows;
cost of food sold = Benning inventory + purchases - ending inventory
From the equation above, an increase in the ending inventory will cause the cost of food sold at the end of the year to appear lower that it really is.
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