generate a continuous and differentiable function f(x) with the following properties: f(x) is decreasing at x=−5 f(x) has a local minimum at x=−3 f(x) has a local maximum at x=3
Answers
Answer:
see details in graph and below
Step-by-step explanation:
There are many ways to generate the function.
We'll generate a function whose first derivative f'(x) satisfies the required conditions, say, a quadratic.
1. f(x) has a local minimum at x = -3, and
2. a local maximum at x = 3
Therefore f'(x) has to cross the x-axis at x = -3 and x=+3.
Furthermore, f'(x) must be increasing at x=-3 and decreasing at x=+3.
f'(x) = -x^2+9
will satisfy the above conditions.
Finally f(x) must be decreasing at x= -5, which implies that f'(-5) must be negative.
Check: f'(-5) = -(-5)^2+9 = -25+9 = -16 < 0 so ok.
f(x) can then be obtained by integrating f'(x) :
f(x) = integral of -x^2+9 = -x^3/3 + 9x = 9x - x^3/3
A graph of f(x) is attached, and is found to satisfy all three conditions.
A function is differentiable at [tex]x = a[/tex], if the function is continuous at [tex]x = a[/tex]. The function that satisfy the given properties is [tex]f(x) = 9x - \frac{x^3}{3} + 3[/tex]
Given that:
The function decreases at [tex]x = -5[/tex] means that: [tex]f(-5) < 0[/tex]
The local minimum at [tex]x = -3[/tex] and local maximum at [tex]x = 3[/tex] means that:
[tex]x = -3[/tex] or [tex]x = 3[/tex]
Equate both equations to 0
[tex]x + 3 = 0[/tex] or [tex]3 - x = 0[/tex]
Multiply both equations to give y'
[tex]y' = (3 - x) \times (x + 3)[/tex]
Open bracket
[tex]y' = 3x + 9 - x^2 - 3x[/tex]
Collect like terms
[tex]y' = 3x - 3x+ 9 - x^2[/tex]
[tex]y' = 9 - x^2[/tex]
Integrate y'
[tex]y = \frac{9x^{0+1}}{0+1} - \frac{x^{2+1}}{2+1} + c[/tex]
[tex]y = \frac{9x^1}{1} - \frac{x^3}{3} + c[/tex]
[tex]y = 9x - \frac{x^3}{3} + c[/tex]
Express as a function
[tex]f(x) = 9x - \frac{x^3}{3} + c[/tex]
[tex]f(-5) < 0[/tex] implies that:
[tex]9\times -5 - \frac{(-5)^3}{3} + c < 0[/tex]
[tex]-45 - \frac{-125}{3} + c < 0[/tex]
[tex]-45 + \frac{125}{3} + c < 0[/tex]
Take LCM
[tex]\frac{-135 + 125}{3} + c < 0[/tex]
[tex]-\frac{10}{3} + c < 0[/tex]
Collect like terms
[tex]c < \frac{10}{3}[/tex]
[tex]c <3.33[/tex]
We can then assume the value of c to be
[tex]c=3[/tex] or any other value less than 3.33
Substitute [tex]c=3[/tex] in [tex]f(x) = 9x - \frac{x^3}{3} + c[/tex]
[tex]f(x) = 9x - \frac{x^3}{3} + 3[/tex]
See attachment for the function of f(x)
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Related Questions
What are the approximate solutions of the graphed function?
Answers
Answer:
x = -2.6, x = 2.6
Step-by-step explanation:
The graph crosses the x-axis at approximately 2.6 and -2.6.
The required approximate solution of the function graphed is x = -2.6 and 2.6.
Given that,
A graph of a function is plotted, and the solution of the function is to be determined.
What are functions?
Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
What is a graph?The graph is a demonstration of curves that gives the relationship between the x and y-axis.
Here, the solution of the function is that value of x where the function terminates to zero, So the given curve terminates to zero at two places at x = -2.6 and x = 2.6 from the observation of the graph.
Thus, the required approximate solution of the function graphed is x = -2.6 and 2.6.
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What is the solution to this system of linear equations?
y-x = 6
y + x = -10
(-2,-8)
(-8.-2)
(6.-10)
(-10.6)
Answers
Answer:
The correct answer is A
Step-by-step explanation:
Answer:
(-8, -2)
Step-by-step explanation:
y-x = 6
y + x = -10
Add the two equations together to eliminate x
y-x = 6
y + x = -10
--------------------
2y = -4
Divide by 2
2y/2 = -4/2
y = -2
Now find x
y+x = -10
-2+x = -10
x = -8
Difference between 5429 and 5907 to the greatest place.
Answers
Given numbers
54295907Inorder to find the difference between them we have to substarct them.
[tex]\\ \sf\longmapsto 5907-5429[/tex]
[tex]\\ \sf\longmapsto 478[/tex]
Answer:
478
Step-by-step explanation:
To solve the problem
5907 -
5429 =
478
please help!! need answer will give brainliest
Answers
Answer:
k(3) is 7 and f(h(15)) is 43
Step-by-step explanation:
(a).
[tex]{ \sf{k(x) = h(x) + g(x)}} \\ { \sf{k(x) = (3 \sqrt{x + 1}) + ( - {x}^{2} + 3x + 1) }} \\ { \sf{k(x) = (3 \sqrt{3 + 1} ) + ( - {3}^{2} + 3(3) + 1) }} \\ { \sf{k(x) = 6 + 1}} \\ { \sf{k(x) = 7}}[/tex]
b).
[tex]{ \sf{f(h(x)) = 4(3 \sqrt{x + 1}) - 5 }} \\ { \sf{f(h(15)) = 4(3 \sqrt{15 + 1}) - 5 }} \\ { \sf{f(h(15)) = 4(3 \sqrt{16} ) - 5}} \\ { \sf{f(h(15)) = 48 - 5}} \\ { \sf{f(h(15)) = 43}}[/tex]
Six people attend the theater together and sit in a row with exactly six seats.
a. How many ways can they be seated together in the row?
b. Suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat?
c. Suppose the six people consist of three married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?
Answers
Answer
A. 720 ways
B. 240 ways
C. 6 ways
There are 720 ways they can be seated in a row together, 120 ways can the people be seated together in a row with the doctor in an aisle seat, and 6 ways can the six be seated together in a row.
What is a permutation?A permutation is defined as a mathematical process that determines the number of different arrangements in a set of objects when the order of the sequential arrangements.
It is assumed that six people will attend the theater together and sit in a row of six.
The following are the various ways they can be seated in a row together:
⇒ 6!
⇒ 6 × 5 × 4 × 3 × 2 × 1
⇒ 720
If the doctor sits in the aisle seat, the remaining 5 persons can sit in the remaining 5 seats 5! ways
The total possibilities are as follows:
⇒ 5 × 4 × 3 × 2 × 1
= 120
Consider that the six persons are made up of three married couples.
In addition to the aforementioned, divide the six chairs into three groups of two seats each.
There is one choice in each block for the husband to be placed on the left and the wife to be positioned on the right and the 3 couples can be seated in the 3 blocks in 3! ways.
⇒ 3 × 2 × 1
The required answer is 6.
Thus, there are 720 ways they can be seated in a row together, 120 ways can the people be seated together in a row with the doctor in an aisle seat, and 6 ways can the six be seated together in a row.
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how many hours are there from 10:30 on Monday to 11:30 on Tuesday
Answers
Answer:
25 hours.
Step-by-step explanation:
We are to find the number of hours from:
Monday at 10:30
to
Tuesday at 11:30
Since the question does not state whether the times are in A.M / P.M ,
There are 24 hours upto 10:30 on Tuesday.
Adding 1 hour gives 11:30
So the answer = 24 + 1 = 25 hrs
Answer:
25 hours
Step-by-step explanation:
1 day=24 hours
11.30-10.30=1 hour
----------------------------
add the total 25 hours
Let X denote the day she gets enrolled in her first class and let Y denote the day she gets enrolled in both the classes. What is the distribution of X
Answers
Answer:
X is uniformly distributed.
Step-by-step explanation:
Uniform Distribution:
This is the type of distribution where all outcome of a certain event have equal likeliness of occurrence.
Example of Uniform Distribution is - tossing a coin. The probability of getting a head is the same as the probability of getting a tail. The have equal likeliness of occurrence.
Diana paints 150 fence posts and chuck paints 130 fence posts. Diana paints 10 more fence posts than chuck. How many fence posts does chuck paint per hour?
Answers
Complete Question
The complete question is shown on the first uploaded image
Answer:
Step-by-step explanation:
From the question we are told that
The relationship is [tex]\frac{150 }{d} = \frac{130}{c}[/tex]
The number of fence post painted by chuck is [tex]l = 130[/tex]
The number of fence post painted by Diana is [tex]k = 150[/tex]
can paint 10 fences more than chuck so let say the of fence painted in an hour by chuck is [tex]g[/tex]
Then the number of fence post painted by Diana in one hour is
[tex]f = g+ 10[/tex]
So
[tex]\frac{150 }{ g + 10 } = \frac{130}{g}[/tex]
[tex]130 g + 1300 = 150g[/tex]
[tex]g = 65 \ m[/tex]
Tại sao số mắt xích không được là bội số của số răng đĩa xích
Answers
Answer:
Nên sử dụng số răng lẻ cho đĩa xích dẫn động kết hợp với số lượng mắt xích chẵn để mài mòn đồng đều trên răng và trục lăn. Trong trường hợp này, một răng cụ thể của bánh xích không tiếp xúc với một mắt xích cụ thể của xích trong mỗi lần quay.
Step-by-step explanation:
Translated
It is preferable to use an odd number of teeth for the driving sprocket in combination with an even number of chain links for uniform wear and tear on the teeth and rollers. In this case, a particular tooth of the sprocket wheel does not come in contact with a particular link of the chain for every rotation.
which statement correctly describes the relation between the variable in the equation C = nd
Answers
Answer:
nd is c
Step-by-step explanation:
the number of times n a ball bounces
Choose the correct answer below.
O rational numbers
O integers
o irrational numbers
O natural numbers
Answers
The number of times n a ball bounces is a natural number
How to determine the category of the statement?The mathematical statement is given as:
The number of times n a ball bounces
A ball cannot bounce in decimal times or negative times
Also, it can not bounce in 0 times
This means that the number of bounce is a positive integer
A positive integer is also a natural number
Hence, the number of times n a ball bounces is a natural number
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A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste.
Which of these is an example of descriptive statistics?
a.)
40% of the people in the city where the bakery is located like the taste of the cookie.
b.)
40% of the surveyed customers like the taste of the cookie.
c.)
40% of all the bakery's customers like the taste of the cookie.
d.)
40% of all people like the taste of the cookie.
Answers
Answer:[ 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics. ]
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste. [ 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics. ]
An example of the disruptive statistics is 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
We have given that,
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste.
We have to determine the
an example of descriptive statistics
What are the descriptive statistics?
Descriptive statistics is a set of brief descriptive coefficients that summarize a given data set representative of an entire or sample population.
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste. 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
An example of the disruptive statistics is that 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
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The expression 2(l + w) is used to calculate the perimeter of a rectangle, where l is length and w is width. If the length is Fraction 2 over 5 unit and the width is Fraction 1 over 5 unit, what is the perimeter of the rectangle in units?
Fraction 3 over 5 unit
1 unit
1Fraction 1 over 5 units
2Fraction 3 over 5 units
Answers
Answer:
Step-by-step explanation:
L = ⅖ unit
W = ⅕ unit
L+W = ⅗ unit
Perimeter = 2×⅗ = 6/5 units = 1⅕ units
Answer:
L = ⅖ unit
W = ⅕ unit
L+W = ⅗ unit
Perimeter = 2×⅗ = 6/5 units = 1⅕ units
Step-by-step explanation:
GIVING BRAINLIEST!!!!! AND ALL POINTS!!!!!!!!!!!!!!!!!!!
A right rectangular prism is packed with cubes of side length fraction 1 over 4 inch. If the prism is packed with 12 cubes along the length, 8 cubes along the width, and 5 cubes along the height, what is the volume of the prism?
fraction 2 and 3 over 4 cubic inches
fraction 3 and 3 over 4 cubic inches
fraction 7 and 1 over 4 cubic inches
fraction 7 and 1 over 2 cubic inches
Answers
Answer:
7 and 1 over 2 cubic inches ( 7 1/2 in³
Step-by-step explanation:
The height = 1/4 * 5 = 1 1/4 = 1.25
The width = 1/4 * 8 = 2
The length = 1/4 * 12 = 3
Volume = 1.25 * 2 * 3 = 2.5 * 3 = 7.5
0.5 is represented as 1/2
So answer : fraction 7 and 1 over 2 cubic inches or 7 1/2 in³
if my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Consider the given statement: Not all rooms at the hotel have a bed.
For each statement below, decide whether it is a negation of the given statement.
A.) No room at the hotel has a bed YES/NO
B.) Some rooms at the hotel have a bed YES/NO
C.) Some rooms at the hotel do not have a bed YES/NO
D.) All rooms at the hotel have a bed YES/NO
Answers
Answer:
The negation is the OPPOSITE of whatever is said, so in this case, the opposite of "Not all rooms at the hotel have a bed." would be D "All rooms at the hotel have a bed" and A "No room at the hotel has a bed"
The correct answers are A and D
Let me know if this helps!
Answer:
a. no because it states that no rooms have a bed when it states not all rooms at the hotel have a bed
b. yes because while its true some rooms dont have a bed other rooms will
c. yes again some rooms do have a bed and some dont
d. no because we have already established that only some rooms have beds not all
hope this helps
Step-by-step explanation:
PLEASE HELP!! WHOEVER HELPS FIRST AND GETS IT CORRECT GETS BRAILIEST!!
Answers
Answer:
16 rotations
Step-by-step explanation:
500/32 = 15.625
15.625 rounded = 16
hope it helps!
Mark has a collection of 80 coins. There are only nickels and dimes in the collection. The total value of the coins is $5.00. How many dimes does Mark have?
Answers
Answer:
number of nickel = 60
number of dimes = 20
Step-by-step explanation:
1 nickel = 5 cents
1 dimes = 10 cents
$1 = 100 cents
we will use these value to solve the questions
_______________________________
Total no of coins = 80
let the number of nickels be x
let the number of dimes be y
thus,
x+y = 80
y = 80-x equation 2
value of x nickels = 5x
value of y dimes = 10y
Total value of x nickels and y dimes = 5x+10y
The total value of the coins is $5.00
total value of the coins in cents = 5*100 = 500
thus
5x+10y = 500
using y = 80-x from equation 2
5x + 10(80 - x) = 500
5x + 800 - 10x = 500
-5x = 500 - 800 = -300
x = -300/-5 = 60
Thus,
number of nickel = 60
number of dimes = 80-60 = 20
How do u solve A/B + C/D = E
Answers
If 4 pounds of cherries cost $10, what is the unit price
Answers
Answer:
2.5$ per pound
Step-by-step explanation:
The cost of 4 pounds of cherries cost 10$
So to khow the price of 1 pound we must divide 10 by 4.
● 10/4 = 2.5
So 1 pound costs 2.5$
What is the size of each piece in square units?
What is the area of the shaded rectangle in square units?
Answer Asapp plezz
Answers
Answer:
8 pieces are shaded
Size of one square is (1/3)*(1/5)=1/15
The area of shaded region is basically (2/3)*(4/5)=8/15
The Size of one square is 1/15. The area of the shaded region rectangle is 8/15.
What is the area of the rectangle?The area of the rectangle is the product of the length and width of a given rectangle.
The area of the rectangle = length × Width
From the given figure we can see that 8 pieces of the square makes a shaded region.
The area of a square = side x side
[tex]= \dfrac{1}{3}\times\dfrac{1}{5}\\\\=\dfrac{1}{15}[/tex]
Therefore, the Size of one square is 1/15.
The area of the shaded region rectangle = length × Width
[tex]= \dfrac{2}{3}\times\dfrac{4}{5}\\\\=\dfrac{8}{15}[/tex]
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Simplify the equation
(X 2/5) 10
Answers
Answer:
x^4
Step-by-step explanation:
x^(2/5) ^10
We know that a^b^c = a^(b*c)
x^(2/5*10)
x^4
Answer:
[tex]{x}^{4} [/tex]
Step-by-step explanation:
[tex] ({x}^{ \frac{2}{5} } ) ^{10} \\ {x}^{ \frac{2}{5} \times 10} \\ {x}^{ \frac{20}{5} } \\ {x}^{4} [/tex]
round to
237 divide by 4 = (hundredths)
Answers
Hope that helps
Use the figure to name a plane containing point
L.
Answers
A geometric sequence is a sequence of numbers where the next term equals to
the previous term multiplied by a common factor (for example, (3, 6, 12, 24, ...)
is a geometric sequence with the first term ”3” and the common factor ”2”). If
the 5th term of a geometric sequence is 24 and the 7th term is 144, what is the
first term of the sequence?
(A) 2
(B) 3/2
(C) 2/3
(D) 1/3
(E) 1/4
Answers
Answer:
C
Step-by-step explanation:
Let the first term be a and the common ratio be r.
ATQ, ar^4=24 and ar^6=144, r=sqrt(6) and a=24/(sqrt(6))^2=24/36=2/3
Find the greatest number of students to whom 45 soccer balls and 60 tennis balls can be equally divided.
Answers
Answer:
15 studentsStep-by-step explanation:
Find GCF of 45 and 60:
45 = 3*3*560 = 2*2*3*5GCF(45,60) = 3*5 = 15
Find Greatest common multiple or GCF
[tex]\\ \sf\longmapsto 45=3\times 3\times 5[/tex]
[tex]\\ \sf\longmapsto 60=2\times 2\times 3\times 5[/tex]
Now
[tex]\\ \sf\longmapsto GCF=3\times 5=15[/tex]
Find the total amount in the compound interest account.
$10000 is compounded semiannually at a rate of 9% for 22 years.
(Round to the nearest cent.)
Answers
Answer:
$69,361.23
Step-by-step explanation:
[tex] A = P(1 + \dfrac{r}{n})^{nt} [/tex]
[tex] A = 10000(1 + \dfrac{0.09}{2})^{2 \times 22} [/tex]
[tex]A = 10000(1.045)^{44}[/tex]
[tex] A = 69361.23 [/tex]
Answer: $69,361.23
Solve for x. Question 12 options: A) 8 B) 5 C) 14 D) 10
Answers
Answer:
B) 5
Step-by-step explanation:
Proportions:
8 ⇒ 10
20 ⇒ 5x
5x = 20*10/8
5x = 25
x = 25/5
x = 5
I need to find AB can someone please help me ASAP :(
Answers
Answer:
AB = 11
Step-by-step explanation:
Using the external secant property,
CB*CA = CD*CL (=CT^2 if T is tangent throught C)
substitute values,
7*(7+2x+3) = 9*(9+x+1)
Expand and simplify
70+14x = 90 + 9x
5x = 20
x = 4
=>
AB = 2x+3 = 2(4)+3 = 8+2 = 11
(1-Cota)^2
+(tana-1)^2=4cosec2a(cosec2a-1)
Answers
Answer:
Step-by-step explanation:
(1-CotA)² + (tanA-1)² = 4csc2A(csc2A-1)
To prove this equation we will take the expression given in left hand side and will convert it into the expression given in right hand side of the equation.
L.H.S. = (1-CotA)² + (tanA-1)²
= 1 + Cot²A - 2CotA + 1 + tan²A - 2tanA
= cosec²A - 2CotA + Sec²A - 2tanA
[Since, (1 + Cot²A = cosec²A) and (1 + tan²A = Sec²A)]
= (cosec²A + Sec²A) - 2(CotA + tanA)
= [tex](\frac{1}{\text{SinA}})^{2}+(\frac{1}{CosA} )^{2}-2\text{(tanA}+\frac{1}{\text{tanA}})}[/tex]
= [tex]\frac{1}{(\text{SinA.CosA})^2}-2(\frac{tan^2A+1}{tanA} )[/tex]
= [tex]\frac{4}{\text{(Sin2A})^{2}}-4(\frac{1}{\text{Sin2A}} )[/tex]
[Since 2SinA.CosA = Sin2A and [tex]\frac{2(\text{tanA})}{1+\text{tan}^{2}A}=\text{Sin2A}[/tex]]
= 4Cosec²2A - 4Cosec2A
= 4Cosec2A(Cosec2A - 1)
= R.H.S. (Right hand side)
Hence the equation is proved.
20 POINTS! You are planning to use a ceramic tile design in your new bathroom. The tiles are equilateral triangles. You decide to arrange the tiles in a hexagonal shape as shown. If the side of each tile measures 9 centimeters, what will be the exact area of each hexagonal shape?
Answers
Answer:
210.33 cm^2
Step-by-step explanation:
We know that 6 equilateral triangles makes one hexagon.
Also, an equilateral triangle has all its sides equal.
If the tile of each side of the triangular tile measure 9 cm, then the height of the triangular tiles can be gotten using Pythagoras's Theorem.
The triangle formed by each tile can be split along its height, into two right angle triangles with base (adjacent) 4.5 cm and slant side (hypotenuse) of 9 cm. The height (opposite) is calculated as,
From Pythagoras's theorem,
[tex]hyp^{2} = adj^{2} + opp^{2}[/tex]
substituting, we have
[tex]9^{2} = 4.5^{2} + opp^{2}[/tex]
81 = 20.25 + [tex]opp^{2}[/tex]
[tex]opp^{2}[/tex] = 81 - 20.25 = 60.75
opp = [tex]\sqrt{60.75}[/tex] = 7.79 cm this is the height of the right angle triangle, and also the height of the equilateral triangular tiles.
The area of a triangle = [tex]\frac{1}{2} bh[/tex]
where b is the base = 9 cm
h is the height = 7.79 cm
substituting, we have
area = [tex]\frac{1}{2}[/tex] x 9 x 7.79 = 35.055 cm^2
Area of the hexagon that will be formed = 6 x area of the triangular tiles
==> 6 x 35.055 cm^2 = 210.33 cm^2