For the function f(x) = 2x^3 + 3x^2 +1:
a. Using Calculus and a sign chart, find the intervals on which f(x) is increasing and decreasing, and identify any local extrema. Give intervals in interval notation and local extrema as ordered pair(s).
b. Using Calculus and a sign chart, determine where f(x) is concave up and concave down, and locate any inflection points. Give intervals in interval notation and inflection points as ordered pair(s).
Answers
a. Intervals of increase: (-1, 0) and (0, ∞ Intervals of decrease: (-∞, - Local minimum: (-1, 2) b. Interval of concave up: (-1/2, ∞) Interval of concave down: (-∞, -1/2 Inflection point: (-1/2, 5/4)
To find the intervals on which the function is increasing or decreasing and to identify any local extrema, we need to find the derivative of the function and analyze its sign.
a. First, let's find the derivative of f(x) by applying the power rule:
f'(x) = 6x^2 + 6x
Now, we can create a sign chart to determine the intervals of increase and decrease and identify local extrema.
Sign chart for f'(x):
Interval | f'(x)
----------------
x < -1 | (-)
-1 < x < 0 | (+)
0 < x | (+)
From the sign chart, we can conclude the following:
- f(x) is decreasing for x < -1.
- f(x) is increasing for -1 < x < 0.
- f(x) is increasing for x > 0.
To identify local extrema, we need to find the critical points by setting the derivative equal to zero and solving for x:
6x^2 + 6x = 0
6x(x + 1) = 0
This equation is satisfied when x = 0 or x = -1. Therefore, the critical points are x = 0 and x = -1.
Now, we can evaluate f(x) at these critical points and the endpoints of the intervals to determine the local extrema:
f(-∞) = lim(x->-∞) f(x) = -∞
f(-1) = 2(-1)^3 + 3(-1)^2 + 1 = -2 + 3 + 1 = 2
f(0) = 2(0)^3 + 3(0)^2 + 1 = 1
f(∞) = lim(x->∞) f(x) = +∞
Therefore, the local extrema are:
- Local minimum at (-1, 2)
b. To determine where f(x) is concave up or concave down and locate any inflection points, we need to analyze the second derivative of f(x).
Taking the derivative of f'(x), we find:
f''(x) = 12x + 6
Now, let's create a sign chart for f''(x):
Sign chart for f''(x):
Interval | f''(x)
----------------
x < -1/2 | (-)
x > -1/2 | (+)
From the sign chart, we can conclude the following:
- f(x) is concave down for x < -1/2.
- f(x) is concave up for x > -1/2.
To find the inflection point(s), we need to find where the second derivative changes sign, which is at x = -1/2.
Evaluating f(x) at x = -1/2:
f(-1/2) = 2(-1/2)^3 + 3(-1/2)^2 + 1 = -1/4 + 3/4 + 1 = 5/4
Therefore, the inflection point is:
- Inflection point at (-1/2, 5/4)
In summary:
a. Intervals of increase: (-1, 0) and (0, ∞)
Intervals of decrease: (-∞, -1)
Local minimum: (-1, 2)
b. Interval of concave up: (-1/2, ∞)
Interval of concave down: (-∞, -1/2)
Inflection point: (-1/2, 5/4)
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Related Questions
Which polynomial function could be represented by the graph below?
On a coordinate plane, a parabola opens up. It goes through (negative 2, 0), has a vertex at (negative 0.5, negative 4.5), and goes through (1, 0).
f(x) = x2 + x – 2
f(x) = 2x2 + 2x – 4
f(x) = x2 – x – 2
f(x) = 2x2 –2x – 4
Answers
Answer:
B! just took it!
Step-by-step explanation:
Answer:
B. f(x) = 2x2 + 2x – 4
Step-by-step explanation:
it's lit!!!
Please answer this for me! I put the picture in.
Answers
Answer:
A
Step-by-step explanation:
start by subtracting 23 from both sides
then you have the equation 14x = 14x-46 which you can't solve
hi cn i pleas have help on this question
Answers
Answer:
396 cm squared
Step-by-step explanation:
10*27=270 cm squared
9*14=126 cm squared
Answer
i think its 486cm squared
Step-by-step explanation:
9*24=216cm
10*27=270cm
216cm+270cm=486cm squared
but im exactly sure
(75 POINTS) BRAINLIEST Graph g(x), where f(x) = 2x − 5 and g(x) = f(x + 1).
Answers
Answer:
Graph g(x) = 2x - 3
Step-by-step explanation:
Plug in (x+1) to f(x) = 2x - 5:
g(x) = 2(x+1) - 5 = 2x + 2 - 5
g(x) = 2x - 3
Find the equation of the line.
Use exact numbers.
Y=?x+?
Answers
Answer:
y=.75x-2
Step-by-step explanation:
4,1
8,4
3/4 or .75x
to check answer:
(.75*8)=6
6-2
4
The equation of the line shown in the graph is y = 4x / 5 -2.
What is the slope of the line?The slope of the line is a tangent angle made by line with horizontal. i.e. m =tanx where x in degrees.
Here,
Points located on the graph that lies on the line are (0, -2), and (2.5, 0)
The slope of the line is given as,
m = (y₂ - y₁) / (x₂ - x₁)
Substitute the value in the above equation,
m = 0+ 2 / 2.5 -0 = 20/25
m = 4 / 5
The equation of the line is given as
y - y₁ = m (x - x₁)
Substitute the one point (-2, 0) and m in the above equation,
y + 2 = 4/5 (x - 0)
y = 4x / 5 -2
Thus, the equation of the line shown in the graph is y = 4x / 5 -2.
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< Quadrilateral L' is the image of quadrilateral L under a dilation. What is the center of the dilation? ( pls help i need it asap )
Answers
The center of the dilation from the quadrilateral WXYZ to quadrilateral W'X'Y'Z' is point A.
What is a dilation?Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.
Given that, quadrilateral L' is the image of quadrilateral L under a dilation.
The origin is often used as the center of dilation.
Connect all the points from quadrilateral WXYZ and quadrilateral W'X'Y'Z' to point A.
Therefore, the center of the dilation is point A.
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What is the algebraic expression for half of a number?
Answers
The algebraic expression for half of a number is x/2.
What is the algebraic expression for half of a number?When we are working in algebra and we want to represent "a number", we use a variable for it.
We do this because "a number" can be any real number.
For example, we can say that a number is represented by the variable x.
Now, to write half of a number, we just need to divide our variable by 2, then we will get:
x/2
That is the algebraic expression.
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Scenario A. The manager at Dunder-Mifflin Paper Company interested in understanding how a company's employee benefits influence employee satisfaction. In 2020 the company implemented a new benefits package that included optional benefits such as childcare, eldercare, and retirement packages. The manager compares the employee satisfaction ratings from before and after the new benefits package was implemented.
1. What is the independent variable for Scenario A?
a. The employee benefits package
b. The work from home policy
c. Employee productivity
d. The employees at the company
e. The office layout (floorplan)
Answers
The independent variable for Scenario A is given as follows:
a. The employee benefits package.
What are dependent and independent variables?In the case of a relation, we have that the independent and dependent variables are defined by the standard presented as follows:
The independent variable is the input of the relation.The dependent variable is the output of the relation.In the context of this problem, we have that the input and the output of the relation are given as follows:
Input: Employee benefits package.Output: Employee satisfaction.Hence the independent variable for Scenario A is given as follows:
a. The employee benefits package.
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The sum of numbers is 28. the difference of two numbers is -2. find the numbers
Answers
Answer:
x=13 and y=15
Step-by-step explanation:
Let the 2 numbers be x and y
ATQ,
x+y = 28 ------ eqn.1
x-y=-2 -------- eqn.2
Adding both the equations:
x+y=28
+x-y=-2
_______
2x = 26
Therefore x=13
Now substituting the value of x in eqn. 1
x+y=28
13+y=28
Therefore y=28-13
=15
One of your friends gives you $10 for a charity walkathon. Another friend gives you an amount per mile. After 5 miles, you have raised $13.50 total. Write an equation in slope-intercept form that represents the amount y of money you have raised after x miles.
y=
Answers
Answer:
y = 0.70x- 10.00
Step-by-step explanation:
-- Since one friend gave you 10.00 ,subtractthis from the 13.50 to give you the amount earned by walking the 5 miles
-- 13.5-10.00 = $3.50
-- Since you walked 5 miles, divide by 5 to give you 0.70 per mile earned.
So, an equation that represents the amount of money earned is:
y = 0.70x- 10.00 . This is in slope-intercept form; the slope is $0.70 and represents the amount of money earned per mile and the intercept is $10.00 and represents the initial amount of money that you were given.
after polygon stuvw is reflected across the y-axis, what are the coordinates of s', the image of point s after the transformation?
Answers
After polygon STUV is reflected across the y-axis, the coordinates of point S' are (-2, 3).
Reflecting a polygon across the y-axis is a type of transformation in geometry that involves flipping the shape over the y-axis. When reflecting a polygon across the y-axis, the x-coordinates of all the points in the shape are multiplied by -1, while the y-coordinates remain the same.
In the case of polygon STUV, point S has coordinates (2, 3). When reflected across the y-axis, the x-coordinate becomes -2, while the y-coordinate remains 3. Therefore, the image of point S after the transformation, denoted as S', has coordinates (-2, 3).
It's important to note that reflecting a polygon across the y-axis is just one of many possible transformations in geometry. Other common transformations include translations, rotations, and dilations. These transformations can be used to explore the properties of shapes and to solve problems in geometry.
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I need the formula for this graph
Answers
Answer:
y=\(\frac{1}{3}\)x+\(\frac{1}{3}\) /
Step-by-step explanation:
For each of the last three years, your landscaping business has made a profit of $1,200; $2,100; and $3,400. How much total profit did your business make in the last 3 years?
O $5,670
O $6,070
O $6,700
O $7,600
O $7,650
Answers
Answer:
The answer would be $6,700
What type of figure is shown?
The base of the figure is a ___________
A: Prism
B: Rectangle (wrong)
C: Triangle
Answers
Answer:
1. triange 2. trangle prism
Step-by-step explanation:
Solve the equation
56 ÷ (-8)= ?
Answers
Answer:
-7
Step-by-step explanation:
Because for dividing one positive and one negative number, the rule is that the answer is going to be negative. So, what you can do is just do 56÷8 which equals 7 then add the negative sign to make it -7.
5^=1 ÷ 652 tell the answer
Answers
Answer:
n = - 4
Step-by-step explanation:
note that 625 = \(5^{4}\) , then
\(5^{n}\) = \(\frac{1}{5^{4} }\) = \(5^{-4}\)
Since the bases on both sides are equal, both 5 then equate the exponents
n = - 4
Answer:
n=-4
Step-by-step explanation:
Note: It is not given what to find so that i am finding the value of n
Given :
\(5^{n} =\frac{1}{625} \\5^n=\frac{1}{5^4} \\5^n=5^{-4} \\So,\\n=-4\)
Therefore,
n=-4
Write the system first as a vector equation and then as a matrix equation. 5x1 + x2 - 3x3 = 8 2x2 + 4x3 = 0
Answers
The system can be written as a vector equation as [5, 1, -3] [x1, x2, x3]^T = [8, 0]^T and as a matrix equation as AX = B, where A = [5 1 -3; 0 2 4], X = [x1; x2; x3], and B = [8; 0].
To write the given system as a vector equation, we group the variables and the constants into vectors and write the equations in a matrix form. Thus, the system can be written as [5x1 + x2 - 3x3; 2x2 + 4x3] = [8; 0], which is a vector equation.
To write the system as a matrix equation, we can write the coefficients of the variables in a matrix A, the variables in a vector X, and the constants in a vector B. Thus, the system can be written as AX = B, where A = [5 1 -3; 0 2 4], X = [x1; x2; x3], and B = [8; 0].
We can then solve for X by finding the inverse of A and multiplying both sides of the equation by it.
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3. Find \( y^{\prime} \) for the following implicit function \( y^{2}-x^{2} y=-2 \)
Answers
The derivative \(\( y' \)\) of the implicit function \(\( y^2 - xy = -2 \)\) is 0, indicating a constant slope with no change in relation to \(\( x \)\).
To find \(\( y' \)\)for the implicit function \(\( y^2 - xy = -2 \)\), we can differentiate both sides of the equation with respect to \(\( x \)\) using the chain rule. Let's go step by step:
Differentiating \(\( y^2 \)\) with respect to \(\( x \)\) using the chain rule:
\(\[\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}\]\)
Differentiating \(\( xy \)\) with respect to \(\( x \)\) using the product rule:
\(\[\frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + y \cdot \frac{dx}{dx} = x \cdot \frac{dy}{dx} + y\]\)
Differentiating the constant term (-2) with respect to \(\( x \)\) gives us zero since it's a constant.
So, the differentiation of the entire equation is:
\(\[2y \cdot \frac{dy}{dx} - (x \cdot \frac{dy}{dx} + y) = 0\]\)
Now, let's rearrange the terms:
\(\[(2y - y) \cdot \frac{dy}{dx} - x \cdot \frac{dy}{dx} = 0\]\)
Simplifying further:
\(\[y \cdot \frac{dy}{dx}\) \(- x \cdot \frac{dy}{dx} = 0\]\)
Factoring out:
\(\[(\frac{dy}{dx})(y - x) = 0 \]\)
Finally, solving:
\(\[\frac{dy}{dx} = \frac{0}{y - x} = 0\]\)
Therefore, the derivative \(\( y' \)\) of the given implicit function is 0.
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ILL MARK BRAINLIEST TO FIRST PERSON THAT ANSWERS CORRECTLY!
Annita baby sits at a rate of $7 per hour. Identify the type of relationship this represents. Explain your answer.
Answers
Answer: The relationship is proportional.
Step-by-step explanation: Because the baby sitters work corresponds to their hourly pay. (I think)
a group of tourists on a weeklong bus tour of a european country experienced an outbreak of norovirus. the group had followed a consistent meal time pattern: each morning they had breakfast together in whichever hotel they had stayed from 6:00 a.m. to 7:00 a.m., stopped for lunch from 1:00 p.m. to 2:00 p.m., then had dinner together either at the next hotel or at a restaurant at about 7:00 p.m. the incubation period for norovirus is about 24-48 hours, with a median of about 33 hours. on which day and at which meal was exposure most likely?
Answers
Answer:
Step-by-step explanation:
Subtracting 24 hours (the minimum incubation period) from the time of onset of the first case puts you in the April 20 Dinner interval. Subtracting 33 hours from the median case (which occurred in the 4-8 AM interval) on April 22), puts you in the April 20 4-7 pm interval, near both lunch and dinner that day. While the minimum method points to dinner on April 20, thorough investigators would probably investigate possible exposures at lunch that day, too.
Write each expression in the correct column to show whether the quotient is less than, greater than, or equal to 1
Answers
The expressions based on the information given are illustrated below
How to calculate the expression?3/4 ÷ 1/2 = 1.5 = quotient is grater than 1
1/2 ÷ 3/4 = 2/3 = quotient is less than 1
2/9 ÷ 1/27 = 6 = quotient is grater than 1
5/3 ÷ 20/6 = 1/2 = quotient is less than 1
4/3 ÷ 3/5 = 20/9 = quotient is grater than 1
19/8 ÷ 2 3/8 = 1 = quotient is equal to 1.
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the value of a car is $20,000. it loses 10.3% of its value each year. write an exponential function to determine the value of the car in t years.
Answers
To model the decrease in the value of the car over time, we can use an exponential function of the form:
V(t) = V(0) * e^(-rt)
where:
V(0) is the initial value of the car (in this case, $20,000).
r is the annual rate of depreciation, expressed as a decimal (in this case, 0.103).
t is the number of years since the car was purchased.
Plugging in the given values, we get:
V(t) = $20,000 * e^(-0.103t)
This is the exponential function that models the value of the car in t years.
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please help me lol!!!!!
Answers
Answer:
It increases
Step-by-step explanation:
PLZZ mark brainliest!!!
if s is a linearly dependent set, then each vector is a linear combination of the other vectors in s.
a. True
b. False
Answers
The statement "if s is a linearly dependent set, then each vector is a linear combination of the other vectors in s" is false.
What is the linearly dependent set?A vector set's linear independence is an important attribute. If no vector in a set can be written as a linear combination of the other vectors in the set, the set is said to be linearly independent. The set is said to be linearly dependent if any of the vectors can be represented as a linear combination of the others.
If S is a linearly dependent set, then each vector in S is a linear combination of the others.
This is false.
For example, v₁ = (1,0), v₂ = (2,0), and v₃ = (1,1). Then v₂ = 2v₁ but v₃ is not a linear combination of v₁ and v₂, since it is not a multiple of v₁. But 2v₁ - 1v₂ + 0 v₃ = 0.
If an indexed set of vector S is linear dependent, then it is only necessary that one of the vectors is in the set.
Hence, the correct answer would be option (B).
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what is -9-2(7-1) simplified??? please i really need help FAST
Answers
Answer:
-21
Step-by-step explanation:
1. The base of a rectangular prism has an area of 20 square inches. If the prism has an altitude of 8 inches, determine and state the volume of the prism, in cubic inches?
Answers
Answer:
160 in^3
Step-by-step explanation:
To find the volume, you do the base x height. If the base if 20 in^2 and the altitude is 8 in, then the volume is 20 * 8 in^3
20 * 8 = 160
Zoom if you have to see more clearly ! Answer #4 , #5 , & #6 for me !! Show proof for all please ! I'll be giving brainliest and report wrong answers or answers that are correct but no proof !
Answers
Answer:
N/A; 32; -8
Step-by-step explanation:
4) You have to graph this equation... basically pick two points on the line and connect them. I'd do (0,4) and (4,1) because they're the easiest
5) Just plug -2 into the equation! -3x³+2x², so when you substitute x for -2, you get -3*(-2)³+2*(-2)²=-3*(-8)+2*4=24+8=32
6) -6m+6+6=4-7m, so m=-8
need help on this question
Answers
-2/3 x 3/5 + 5/2 - 3/5 x 1/6
Answers
Answer:
sorry im on a time lapse, so the answer is 2 1/2. hope it helps
Step-by-step explanation:
Answer:
Hi above picture is the solution for your question.
Please mark it the brainliest answer
39 MHF 4UB Unit III Workbook 4. Applications [5 marks] A piston in a large factory engine moves up and down in a cylinder. The height, h centimetres, of the piston at t seconds is given by the function h(t)=120sin at +200. a) Amplitude = period = b) What are the maximum and minimum heights of the piston? c) How many complete cycles does the piston make in 30 min.?
Answers
Max height = A + D = 120 + 200 = 320 cm. Min height = D - A = 200 - 120 = 80 cm. The period of the function can be determined using the formula T = 2π/B, where B is the coefficient of t in the function.
a) To find the amplitude and period of the function, we need to identify the values of "a" in the given function h(t) = 120sin(at) + 200.
The general form of a sinusoidal function is h(t) = A×sin(Bt + C) + D, where:
A represents the amplitude,
B determines the period (T = 2π/B),
C indicates any phase shift, and
D represents a vertical shift.
In the given function h(t) = 120sin(at) + 200, we can see that the coefficient of t is "a." Therefore, the value of "a" represents B in the general form of a sinusoidal function.
Since the given function is h(t) = 120sin(at) + 200, we can deduce that the value of "a" determines the period of the function.
b) To determine the maximum and minimum heights of the piston, we need to find the amplitude of the function. The amplitude (A) represents the maximum displacement from the mean position.
In the given function h(t) = 120sin(at) + 200, we can observe that the amplitude (A) is equal to 120.
The maximum height is given by the sum of the amplitude and the vertical shift (D): Max height = A + D = 120 + 200 = 320 cm.
The minimum height is given by the difference between the amplitude and the vertical shift (D): Min height = D - A = 200 - 120 = 80 cm.
c) The period of the function can be determined using the formula T = 2π/B, where B is the coefficient of t in the function.
Since B = a, we need to find the value of "a" to determine the period. Unfortunately, the value of "a" is not provided in the question. Please check if there is any missing information or additional context that can help us find the value of "a" to calculate the number of complete cycles in 30 minutes.
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