Find the local maximum and minimum values of f using both the first and second derivative tests f(x) = x2 / (x - 1). Summary: The local maximum and minimum values of f(x) = x2 / (x - 1) using both the first and second derivative tests is at x = 0 and x = 2.
Answers
The value of local maximum and local minimum for the function f(x) = x^2/(x -1 ) is equal to f(0) = 0 at x = 0 and f(2) = 4 at x = 4 respectively.
Local maximum and minimum values of the function
f(x) = x^2 / (x - 1),
Use both the first and second derivative tests.
First, let's find the critical points of the function,
By setting its derivative equal to zero and solving for x,
f'(x) = [2x(x - 1) - x^2] / (x - 1)^2
⇒ [2x(x - 1) - x^2] / (x - 1)^2 = 0
Simplifying this expression, we get,
x(x - 2) = 0
This gives us two critical points,
x = 0 and x = 2.
These critical points correspond to local maxima, local minima, or neither.
Use the second derivative test,
f''(x) = [2(x - 1)^2 - 2x(x - 1) + 2x^2] / (x - 1)^3
At x = 0, we have,
f''(0) = 2 / (-1)^3
= -2
Since the second derivative is negative at x = 0, this critical point corresponds to a local maximum.
f(0) = 0^2/ (0 -1 )
= 0
At x = 2, we have,
f''(2) = 2 / 1^3
= 2
Since the second derivative is positive at x = 2, this critical point corresponds to a local minimum.
f(2) = 2^2/ (2 - 1)
= 4
Therefore, at x = 0, the local maximum value is f(0) = 0, and at x = 2, the local minimum value is f(2) = 4.
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Related Questions
determine if the sequence is arithmetic or geometric and determine the common difference / ratio in simplest form 15,10,5
Answers
Answer:
The sequence is arithmetic progression and the common difference is -5
Step-by-step explanation:
AP = a + (n -1)d
a = first number = 15
d = common difference
Using the second term
10 = 15 + (2-1) d
10 = 15 + (1)d
10 = 15 + d
10-15 = d
d = -5
Using the third term
5 = 15 + (3-1) d
5 = 15 + (2)d
5 = 15 + 2d
5-15 = 2d
-10 = 2d
-10/2 = d
-5 = d
For the Geometric Progression,
GP = ar^n-1
a = first number
r = common ratio
Using the second number of the series
10 = 15 × r^2-1
10/15 = r^1
-2/3 = r
Using the second number of the series
5 = 15 × r^3-1
5/15 = r^2
-1/3 = r^2
r = square root of -1/3
Conclusion: From the above calculations, the arithmetic progression has a constant common difference of -5, and the geometric progression does not have a constant common ratio. Hence, the sequence is an arithmetic one
An arithmetic sequence is a sequence where the difference between each consecutive term is the same.
The sequence 15, 10, 5 is an arithmetic sequence.
The common difference is 5.
What is an arithmetic sequence?It is a sequence where the difference between each consecutive term is the same.
Example:
2, 4, 6, 8, 10 is an arithmetic sequence.
We have,
15, 10, 5
This is an arithmetic sequence.
The common difference.
= 15 - 10
= 5 _____(A)
Or
The common difference.
= 10 - 5
= 5 ______(B)
From (A) and (B),
The common difference is the same.
The sequence is an arithmetic sequence.
Now,
The common ratio.
= 10/15
= 2/3 ____(1)
Or
The common ratio.
= 5/10
= 1/2 _____(2)
From (1 )and (2),
The common ratio is not the same
The sequence is not a geometric sequence.
Thus,
The sequence 15, 10, 5 is an arithmetic sequence.
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Solve this system of equations using the
elimination method.
Answers
Answer:
4x-3y=-10 ans C
2x+3y=4
Step-by-step explanation:
4x-3y=-10--------eqn 1
2x+3y=4----------eqn 2
multiply eqn 1 by 2 and eqn 2 by 4
8x-6y=-20
8x+12y=16
subtract eqn 2 from 1
8x-8x ,12-(-6y)=16-(-20)
18y=36
divide both sides by 18
y=2
substitute y=2 in eqn 1
4x-3(2)=-10
4x-6=-10
4x=-10 + 6
4x= -4
divide both sides by 4
x= -1
The circumference of the hub cap of a tire is 74.79 centimeters. Find the area of this hub cap. Use 3.14 for pi . Use pencil and paper. If the circumference of the hub cap were smaller, explain how this would change the area of the hub cap.
Answers
The area of the hub cap with a circumference of 4.79 centimeters is 445.32 cm².
What is the area of the hub cap?The area and circumference of a circle is expressed as;
A = πr²
C = 2πr
Where r is the radius.
Given the data in the question;
Circumference C = 74.79 cmConstant pi π = 3.14radius r = ?Area A = ?To determine the area of hub cap, first find the radius.
C = 2πr
74.79 = 6.28 × r
r = 74.79 / 6.28
r = 11.909 cm
Now, the area of of the cap will be;
A = πr²
A = 3.14 × ( 11.909 cm )²
A = 445.32 cm²
Therefore, the area of the cap is 445.32 cm².
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A new car is bought for $100,000. If the annual depreciation is 10%, find the value of the car after 3 years.
Remaining Amount = 100,000(1 -0.1)3
Answers
The value of the car after 3 years is $79, 000
How to determine the valueTo determine the value, we have to use the formula;
Value = Initial value × (1 - Depreciation rate)ⁿ
Substitute the values given, we get;
Remaining value after 3 years = $100,000 × (1 - 0.10)³
Expand the bracket, we get;
Value after 3 years = $100,000 × (0.90)³
Find the cube value and substitute, we have;
Value after 3 years = $100,000 × 0.729
Multiply the values, we have;
Value after 3 years = $72,900
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Ms. Baetz will receive a 25 % discount off her cellphone bill . Her monthly cell phone plan charges a flat monthly fee $30.00 plus $0.10 per minute . If m.s baetz talks on her call phone for 200 minutes during the first moth of cell phone service , what will be the amount of her first cell phone bill ?
A. 50.00
B. 37.50
C. 42.50
D. 25.00
Answers
Answer:
$50
Step-by-step explanation:
A spherical balloon has a radius of 165 mm. How much air was used to fill this balloon?
Answers
Step-by-step explanation:
To calculate the amount of air used to fill a spherical balloon, we need to determine its volume. The volume of a sphere can be calculated using the formula:
Volume = (4/3) × π × radius³
Given that the radius of the balloon is 165 mm, we can substitute this value into the formula:
Volume = (4/3) × π × (165 mm)³
Calculating this expression will give us the volume of the balloon. However, to provide the answer in a more practical unit, such as liters, we need to convert the volume from cubic millimeters to liters.
1 liter = 1000 cubic centimeters (cc)
1 cubic centimeter (cc) = 1 milliliter (ml)
1 milliliter (ml) = 1 cubic millimeter (mm³)
So, to convert the volume from cubic millimeters to liters, we divide the volume by 1000:
Volume (liters) = Volume (cubic millimeters) / 1000
Now, let's calculate the volume of the balloon and convert it to liters:
Volume (cubic millimeters) = (4/3) × 3.14159 × (165 mm)³
Volume (cubic millimeters) ≈ 4.18879 × (165 mm)³
Volume (liters) = (4.18879 × (165 mm)³) / 1000
Using a calculator to evaluate this expression will give us the volume of the balloon in liters.
Find 8 3/4 ÷ 1 2/7. Write the answer in simplest form.
Answers
Problem: Find 8 3/4 ÷ 1 2/7. Write the answer in the simplest form.
Solution:
\((8+\frac{3}{4}\text{ )}\div(1\text{ + }\frac{2}{3})\)this is equivalent to:
\((\frac{32+3}{4}\text{ )}\div(\text{ }\frac{3+2}{3})\text{ = }(\frac{35}{4}\text{ )}\div(\text{ }\frac{5}{3})\text{ }\)Now, we do cross multiplication:
\(=(\frac{35}{4}\text{ )}\div(\text{ }\frac{5}{3})=\frac{35\text{ x 3}}{5\text{ x 4}}\text{ =}\frac{105}{20}\)then, the correct answer would be:
\(=\frac{105}{20}\)A company wants to increase the 10% peroxide content of its product by adding pure peroxide (100% peroxide). If x liters of pure peroxide are added to 500 liters of its 10% solution,the concentration, C, of the new mixture is given by
C = x+0.1(500) / x+500. x 100
How many liters of pure peroxide should be added to produce a new product that is 36% peroxide?
Answers
The number of liters of pure peroxide that should be added to produce a new product that is 36% peroxide is:203 liters.
Number of litersGiven:
C = x+0.1(500) / x+500. x 100
Concentration, c=36% or.36
Let x represent the number of liters
Hence:
x+0.1(500) / x+500=.36
x+50=.36(x+500)
Simplify
x+50=.36x+180
.64x=130
Divide both side by .64x
x=130/.64
x=203 liters
Therefore the number of liters of pure peroxide that should be added to produce a new product that is 36% peroxide is:203 liters.
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Find the area of the shape
Answers
Hello!
area
= 2*25 + (20 - 2)*(25-8)
= 50cm² + 306cm²
= 356cm²
Kirsty counts the number of peas in a sample of pea pods. Number of pods Number of peas in a pod 4 5 6 7 8 c) 2 10 18 14 6 What was the mean number of peas in a pod?
Answers
The number of pea is 6. By using formulas of mean, median and mode.
What is a mean?A dataset's mean (also known as the arithmetic mean, which differs from the geometric mean) is calculated by dividing the sum of all values by the total number of values. It is frequently referred to as the "average" and is the most widely used measure of central tendency.
Taking pea counts into account,
A pod of peas contains 4, 5, 6, 7 or 8 peas.
Number of pods: 2, 10, 18, 6, and 14
The mode is the number that corresponds to the highest frequency.
The frequency of each number in the list is 1.
Mode is not possible with the information provided.
The middle value, after sorting the numbers in ascending or descending order, is the median if all of the numbers are odd.
The median of the given numbers is.
Replace the values in the formula to get the mean value.
Mean = 4+5+6+7+8/5 = 30/5 = 6
Consequently, based on the quantity of peas provided, the values of the following measure are as follows:
It is impossible to use Peas in a Pod mode.
The median number of peas in a pod is 6.
the typical quantity of peas is 6.
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En el cine, el precio de admisión de un niño es de 6 dólares y de un adulto es de 9,20 dólares . El sábado se vendieron 146 boletos de admisión para un total de ventas de 1135,20 dólares . ¿Cuántos boletos de admisión de niños se vendieron ese día?
Answers
Answer:
65 boletos.
Step-by-step explanation:
Antes de resolverSabemos que el precio para un niño es 6 dólares, y el precio para un adulto es 9,20 dólares. Además, el cine se vendió 146 boletos, y recibió 1135,20 dólares.
El procesoVamos a resolver este problema con un sistema, usando 2 varibles.
No sabemos cuantos boletos de niño el cine vendió, entonces el cine vendió <<x>> boletos de niños. Tampoco sabemos cuantos boletos de adultos vendió, entonces vendió <<y>> boletos de adultos.
En total, vendió 146 boletos; el número de los boletos de niños con el número de los boletos de adultos es 146 boletos.
Usando las variables, x + y = 146 boletos.
Esta es una de las equaciones que necesitamos.
Porque todos los boletos de niños es 6 dólares, si el cine vendió <<x>> boletos, su total de ventas (de boletos de niños) es <<6x>> dólares.
Lo mismo es verdad para los boletos de adultos; todos los boletos de adultos es 9,20 dólares, entonces si vendió <<y>> boletos, el total de ventas de boletos de adultos es <<9,20y>> dólares.
En total, las ventas de los boletos de niños + las ventas de los boletos de adultos es 1135,20 dólares.
O, con variables, es 6x + 9,20y = 1135,20
Nuestra sistema es:
x + y = 146
6x + 9,20y = 1135,20
Ahora necesitamos resolver esta sistema.
Podemos usar la sustitución. Vamos a resolver una de las eqaciones para una variable, y después, usar el valor de esa variable para encontrar la solución para una de las variables, y después, usar esa solución para encontrar la solución de la otra variable.
Podemos mover <<y>> al otro lado.
x + y = 146 => x = 146 - y
Ahora, podemos sustituir <<x>> para 146 - y en 6x + 9,20y = 1135,20
6(146-y) + 9,20y = 1135,20
Multiplica.
876 - 6y + 9,20y = 1135,20
Combina los términos que son los mismos.
876 + 3,20y = 1135,20
Mueva 876 al otro lado.
876 + 3,20y = 1135,20 => 3,20y = 259,20
Divide todo por 3,20.
3,20y/ 3,20 = 259,20/ 3,20
y = 81
Entonces, el cine vendió 81 boletos de adultos.
La respuestaRecuerda, esta (81 boletos de adultos) no es lo que necesitamos. Necesitamos cuantos boletos de niños vendió.
Entonces, podemos substituir <<y>> con 81 en x = 146 - y. Recuerda que <<x>> es cuantos boletos de niños vendió.
x = 146 - y => x = 146 - 81 => 65
Entonces, el cine vendió 65 boletos de niños.
The sales tax rate on a new vehicke is 6.5 percent. If the vehicle is priced at 12,000, how much will you be charged for sales tax?
Answers
If sales tax rate on a new vehicle is 6.5 percent. If the vehicle is priced at 12,000, then 780 will be charged for sales tax
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given,
The sales tax rate on a new vehicle is 6.5 percent.
The price of the vehicle is 12000
We need to find the how much will you be charged for sales tax
Let us convert 6.5 percent to the decimal value by dividing it with 100
6.5/100×12000
0.065×12000
780
Hence, 780 will be charged for sales tax.
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Joseph and Deb deposit $600.00 into a savings account which earns 5% interest compounded
continuously. They want to use the money in the account to go on a trip in 1 year. How much
will they be able to spend?
Round your answer to the nearest cent.
Answers
Answer:
We can use the formula for continuous compound interest to find the balance in Joseph and Deb's savings account after 1 year:
A = Pe^(rt)
where A is the balance, P is the principal (initial deposit), e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate as a decimal, and t is the time in years.
Substituting the given values, we get:
A = $600.00e^(0.05*1)
Using a calculator, we get:
A ≈ $632.57
Therefore, Joseph and Deb will have approximately $632.57 in their savings account after 1 year. They can spend up to this amount on their trip. Rounded to the nearest cent, the answer is $632.57.
- Find the circumference of the circle with the given radius or diameter. Use 3.14.
diameter = 10 cm
A. 15.7 cm
B. 314 cm
C. 78.5 cm
D. 31.4 cm
Answers
Answer:
C = 31.4 cm
Step-by-step explanation:
C = pi * d where d is the diameter
C = 3.14 * 10
C = 31.4 cm
Circumference = pi x diameter
= 3.14 x 10
= 31.4 cm
The answer is D. 31.4 cm.
What is (I^2)+(I^2)+3x if x=I^2?
Show your work.
Answers
Answer:
-5
Step-by-step explanation:
(i^2) = -1 so -1 -1 + 3x = 3x - 2. If x = i^2 = -1, then 3x = -3. -3 - 2 = -5
What is the length of CD?
Answers
Answer:
CD = 15
Step-by-step explanation:
Pre-SolvingWe are given two quadrilaterals: CDEF and GIJH.
We know that CD = 3x - 6, and GI = x + 8.
We want to find the length of CD.
SolvingFinding xNotice how CD and GI both have one tick mark on them.
This indicates that these two are congruent, meaning that they have the same length.
Because of that, CD = GI.
If we substitute 3x - 6 for CD and x + 8 for GI, we get:
3x - 6 = x + 8
Now, we can solve this like a regular equation.
Add 6 to both sides.
3x - 6 = x + 8
+6 +6
_________________
3x = x + 14
Subtract x from both sides.
3x = x + 14
-x -x
_____________-
2x = 14
Divide both sides by 2.
2x = 14
÷2 ÷2
____________
x = 7
Finding the length of CDWe found the value of x.
Now, substitute 7 as x in 3x-6 (the length of CD).
CD = 3x - 6 = 3(7) - 6 = 21 - 6 = 15
Therefore, CD = 15.
please i need help i literally have an hour and 45 minutes to finish this
Answers
Answer:
0.506
0.516
0.609
0.615
Step-by-step explanation:
Plz HELP ASAP (algebra)
Answers
Answer:16
Step-by-step explanation:
Given: 15% of 60 is 9.
so 9 kids are 15 year old.
and 1 other kid is 16 year old.
Thus, there are total of (15+1) kids that are either 15 or 16 year old. you can also draw venn diagram to find that.
A farmer can pick 43 apples in 1 minute. Fill out the table to
see how many apples she can pick in 3 minutes, 5 minutes,
and 10 minutes.
Answers
3x43=120 (calculation)
5x43=215 (calculation)
10x43=430 (calculation)
Minutes Apples
1 43
3 129
5 215
10 430
Hope this helps you!!!!! :D
Answer:
3 mins-129 apples
5 mins-215 apples
10 mins-430 apples
Step-by-step explanation:
3 mins=43x3
5 mins=43x5
10 mins=43x10
Pollsters are concerned about declining levels of cooperation among persons contacted in surveys. A pollster contacts 94 people in the 18-21 age bracket and finds that 80 of them respond and 14 refuse to respond. When 270 people in the 22-29 age bracket are contacted, 240 respond and 30 refuse to respond. Assume that 1 of the 364 people is randomly selected. Find the probability of getting someone in the 22-29 age bracket or someone who agreed to respond.
Report the answer as a percent rounded to one decimal place accuracy. You need not enter the "%" symbol.
Answers
There is a 98.65% chance of finding someone who is between the ages of 22 and 29 or who agrees to answer.
The total number of people who responded is 80+240=320 and the total number of people who refused to respond is 14+30=44.
The probability of getting someone in the 22-29 age bracket or someone who agreed to respond is:
(240/270) + (320/364) - [(240/270)*(320/364)] =0.9865
Multiplying by 100 to convert to percent and rounding to one decimal place, the answer is 98.65%.
Therefore, the probability of getting someone in the 22-29 age bracket or someone who agreed to respond is 98.65%.
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The radius of a spherical ball is increasing at a rate of 2 cm/min. At exactly what rate (in cm^2/min) is the surface area of the ball increasing when the radius is 9 cm?
Answers
Answer:
51
Step-by-step explanation:
The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $600 or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of future weekend guest bills to test the manager's claim. a. Which form of the hypotheses should be used to test the manager's claim? H 0: less than or equal to $600 H a: greater than $600 b. When H 0 cannot be rejected, can we conclude that the manager's claim is wrong? No c. When H 0 can be rejected, can we conclude that the manager's claim is wrong? Yes
Answers
Answer:
H0 : μ ≤ 600
H1 : μ > 600
B.) NO
C.) YES
Step-by-step explanation:
Here, the null hypothesis is the claim, that mean guest bill is $600 or less ; The alternative will be the opposite of the null which is the counterclaim. This ubstge accountants assertion that guest bill has been increasing
Hence,
H0 : μ ≤ 600
H1 : μ > 600
When the null cannot be rejected, that means there is no sufficient evidence to support the alternative hypothesis, Hence, we stick with the manager's claim
When H0 can be rejected, then there is sufficient evidence to support, the accountant's claim. Thus, we go with the accountant's claim and conclude that the manager is wrong.
Solve by using trial and error method : 2x + 3 = 13
Answers
Explanation:
2•5 = 10
10 + 3 = 13
Hopefully this helps you.
two men A and B stand on opposite sides of a 100 metre tower.A observes the angle of elevation of the top of the tower as 45 and B observes it as 30 . how far apart are the men?
Answers
9514 1404 393
Answer:
273 m
Step-by-step explanation:
The tower represents the short side of a 30°-60°-90° right triangle. The distance from the tower represents the long leg, not the hypotenuse, of that triangle. It is √3 times the short side. In this geometry, man B is 100√3 = 173.2 meters from the tower.
The tower represents one of two congruent sides of a 45°-45°-90° right triangle. Man A will be standing 100 meters from the tower.
The distance between A and B is 100 +173.2 = 273.2 meters. The men are about 273 meters apart.
The difference between the areas of the figures is less than 4.
An absolute value inequality that represents this situation is ___
The solution of the inequality is ___
Thank you!!!!!!!!
Answers
Answer:
We know that:
If T = area of the triangle
and R = area of the rectangle:
I T - RI < 4.
Now, we know that:
T = 8*6/2 = 8*3 = 24
R = 4*(x - 4) = 4*x - 16
Then replacing those values, we can write:
I24 - (4*x - 16)I < 4
I40 - 4*xI < 4
Now let's solve it:
First we aim for the first value that is not a solutions, this is when:
I40 - 4*xI = 4
we can write this as:
40 - 4*x = +-4
The first extreme is:
40 - 4*x = +4
x = (40 - 4)/4 = 9
The other extreme is:
40 - 4*x = -4
x = (40 + 4)/4 = 11.
Then the set of solutions is: S = (9, 11)
Higher Order Thinking Morgan read
a thermometer at 7:00 P.M. The
temperature was 16°C. This temperature
was 9°C less than the temperature at
2:00 P.M. The temperature at 2:00 P.M.
was 10°C higher than the temperature at
8:00 A.M. What was the temperature at
8:00 A.M.?
Answers
The temperature at 8:00 A.M. was 15°C.
Using the given information:
1. At 7:00 P.M., the temperature was 16°C.
2. This temperature was 9°C less than the temperature at 2:00 P.M.
We can use this information to find the temperature at 2:00 P.M.:
Temperature at 2:00 P.M. = 16°C (temperature at 7:00 P.M.) + 9°C
Temperature at 2:00 P.M. = 25°C
3. The temperature at 2:00 P.M. was 10°C higher than the temperature at 8:00 A.M.
Now, we can find the temperature at 8:00 A.M.:
Temperature at 8:00 A.M. = 25°C (temperature at 2:00 P.M.) - 10°C
Temperature at 8:00 A.M. = 15°C
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1. Jon waits on tables and is paid $6.75an hour. In a 40-hour period, he earned $310.50 in tips. What were his total earnings?
Answers
Answer:
$617.25
Step-by-step explanation:
$617.25 Because, $310.50(tips) + $306.75(hourly wage)=$617.25
Answer:$580.50
Step-by-step explanation:
:)
The weights of bags of chips are normally distributed with a mean of 180 grams and a standard deviation of 4 grams.
What percent of the bags of chips weigh less than 176 grams?
2.5%
16%
32%
34%
Answers
The percent of the bags of chips weigh less than 176 grams is b 16%
What percent of the bags of chips weigh less than 176 grams?From the question, we have the following parameters that can be used in our computation:
Mean = 180
Standard deviation = 4
Bags of chips weigh less than 176 grams
This means that
x = 176
So, the z-score is
z = (176 - 180)/4
Evaluate
z = -1
The percent of the bags of chips weigh less than 176 grams is represented as
P = P(z < -1)
Using a graphing calculator, we have
Percentage = 0.15866
So, we have
Percentage = 16%
Hence , the percentage is 16%
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how can you say the given set A is a subset of another set B?
Answers
Answer:
Because all elements in set B can be found in set A
Question 3
1 pts
Tim has 74 socks, all in pairs. He has 1 more pair of white socks than black socks. How many individual socks of
each color does he have?
Answers
Answer:
36 individual white socks
38 individual black socks
Step-by-step explanation:
Pairs of socks = 74
He has 1 more pair of white socks than black socks.
Take the colors of the socks to be white and black
Let
Black pairs of socks = x
White pairs of socks = x + 1
Individual socks of each color
A pair of white socks = 2 Individual white socks
= 2white
A pair of black socks = 2 individual black socks
= 2black
Total pairs of socks = black pairs of socks + white pairs of socks
74 = 2white + 2black
74 = 2(x) + 2(x +1)
74 = 2x + 2x + 2
74 = 4x + 2
Subtract 2 from both sides
74 - 2 = 4x
72 = 4x
Divide both sides by 4
x = 72/4
= 18
x = 18
Pairs of white socks = x = 18
A pair of white socks = 2whites
= 2(18)
= 36 individual white socks
Pairs of black socks = x + 1
= 18 + 1
= 19 pairs
A pair of black socks = 2black
= 2(19)
= 38 individual black socks