Answer:
(0, 21)
Step-by-step explanation:
Stationary points occur when the gradient of a graph is zero.
Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.
Differentiate the given function:
[tex]\begin{aligned}y & = x^3+ax^2+b\\\implies \dfrac{\text{d}y}{\text{d}x}&=3 \cdot x^{3-1}+2 \cdot ax^{2-1}+0\\\implies \dfrac{\text{d}y}{\text{d}x}& = 3x^2+2ax\end{aligned}[/tex]
Set the differentiated function to zero:
[tex]\begin{aligned} \dfrac{\text{d}y}{\text{d}x}& =0\\ \implies 3x^2+2ax & = 0 \\ \implies x(3x+2a)&=0\end{aligned}[/tex]
Therefore the stationary points occur when:
[tex]\implies x=0[/tex]
[tex]\implies 3x+2a=0 \implies x=-\dfrac{2}{3}a[/tex]
There is a stationary point at (4, -11), therefore substitute x = 4 into the expression for x and solve for a:
[tex]\begin{aligned}x & = 4\\\implies -\dfrac{2}{3}a & = 4\\a & = -\dfrac{3}{2}(4)\\a & = -6\end{aligned}[/tex]
Substitute the found value of a and the point (4, -11) into the function and solve for b:
[tex]\begin{aligned}y & = x^3 + ax^2 + b\\\implies -11 & = (4)^3+(-6)(4)^2+b\\ -11 & = 64-96+b\\b & = -11-64+96\\ b & = 21\end{aligned}[/tex]
Therefore, the function is:
[tex]\boxed{y=x^3-6x^2+21}[/tex]
The other stationary point is when x = 0. Therefore, to find the coordinates of this point, substitute x = 0 into the function:
[tex]\begin{aligned}y & =x^3-6x^2+21\\x=0\implies y & =(0)^3-6(0)^2+21\\ y & =21 \end{aligned}[/tex]
Therefore, the coordinates of the other stationary point are (0, 21).
To determine if a stationary point is minimum or maximum, differentiate the function again:
[tex]\begin{aligned}y & =x^3-6x^2+21\\\implies \dfrac{\text{d}y}{\text{d}x} & = 3x^2-12x\\\implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\\end{aligned}[/tex]
Substitute the x-coordinate of the stationary point into the second derivative:
[tex]\begin{aligned} \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\x=0 \implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6(0)-12= -12 \\\end{aligned}[/tex]
[tex]\textsf{As $\dfrac{\text{d}^2y}{\text{d}x^2} < 0$, stationary point $(0,21)$ is a maximum.}[/tex]
Differentiation Rules
[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]
Let F(x)=x2-19 and G(x) = 14-x
Find (F/G)(-3)
Value of (F/G)(-3) with the given functions F(x)=x² -19 and G(x) = 14 - x is equal to (-10/17).
As given,
Given functions
F(x)=x² -19
G(x)=14 - x
Substitute the values in the required function:
( F/G )(x)=( x² -19 )/( 14 - x)
Now substitute x= -3
( F/G )( -3) =[ (-3)² -19 ] /[ 14 - (-3) ]
⇒ ( F/G )(-3) =( 9-19) / (14+ 3)
⇒ ( F/G )(-3) =( -10/17)
Therefore, value of (F/G)(-3) with the given functions F(x) = x² -19 and G(x) = 14 - x is equal to (-10/17).
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A train is going on a journey. First it travels east for 50 kilometers. Then it turns north and travels for 22 kilometers. Next it goes west for 50 kilometers. Finally, it travels south for a certain
number of kilometers. At the end of its journey it is back at its original location. How many kilometers did it travel in the southward direction?
Answer:
It traveled south for 22 kilometers.
Step-by-step explanation:
Think of it as a rectangle. Two sides are 50 kilometers (east and west), and the other two sides are 22 kilometers (north and south).
COULD YALL PLEASE HELP ME !!!!!!
Answer:
Adjacent Angles
Step-by-step explanation:
Angles ∠1 and ∠2 are next to each other so adjacent
Find the slope of the line graphed below.
GUYS WHAT HELPPP MEEE
Riley is starting a running plan to train for a race. In the first week of her running
plan, Riley will run 15 miles. The plan calls for Riley to increase her weekly milage by
3 miles every week. If Riley sticks to the plan, how many miles would she be expected
to run during her 5th week of the training plan?
Evaluate x/2-5y+1.5z when x = 10, y = 8, and z = 4
Answer:
-29
Step-by-step explanation:
x/2 - 5y + 1.5z
(10)/2 - 5(8) + 1.5(4)
5 - 40 + 6
-35 + 6
-29
Answer:
=10/2-5×8+1.5×4
=5-40+6.0
=-29
Which mixed number is equivalent to the improper fraction?
41
7
O A. 6-1/
OB. 49/
O C. 59/
OD. 6
PLEASE HELP FAST
Answer:
C -
[tex]5 \frac{6}{7} [/tex]
What is the difference between Earth's highest mountain and its deepest ocean canyon? What is the difference between Mars' highest mountain and its deepest canyon? Which difference is greater? How much greater is it?
earth's difference= 65,233 ft
mars difference= 101,000 ft
greater one is= mars
difference of= 35,767 ft
Jean's bedroom is 12 feet by 10 feet. She has chosen a carpet which costs $30.95 per square yard.
This includes installation.
Answer: i think its 1238 dollars
Step-by-step explanation:
leanna cuts some roses in her front garden and another 20 in her back garden. she uses one quarter of her roses to make a bouquet of a dozen flowers. how many roses did leanna cut in her front yard?
The roses that Leanna cut in her front yard is 28 roses.
How to calculate the value?From the information, she uses one quarter of her roses to make a bouquet of a dozen flowers. Therefore, the total flowers will be:
= 12 ÷ 1/4
= 12 × 4
= 48 flowers
Since she cuts some roses in her front garden and another 20 in her back garden, the number that she cut in the front garden will be:
= 48 - 20
= 28 flowers.
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How can I solve this?
Step-by-step explanation:
by using formula a²-b²=a+b
a²b=a²-b² like this
A technician repaired 18 laptops and
24 desktops last month. What is the
ratio of laptops to total computers?
Therefore, the ratio of the number of laptops to total computers is 2:7
Number of laptops repaired by a technician = 18
Number of desktops repaired by a technician = 24
Total number of computers = no of laptops + no of desktops
= 18 + 24
= 42
We have to determine the ratio of laptops to total computers as
= no of laptops / total no of computers
= 18 / 42
= 6 / 14
= 2 / 7
Therefore, the ratio of the number of laptops to total computers is 2:7
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A company makes pencils. they sell each pencil for $10.
Their revenue is represented by R= 10x
The cost to make the pencils is $3 each with a one time start up cost of $5000.
Their cost is represented by C= 3X+5000
1.Find the profit, P, (P=R-C) when the company sells 1000 pens.
2. Find the number of pens they need to sell to break even ( when R=C)
We have the profit fomula:
P = R - C
= $10.x - ($3.x + 1000)
= $7.x - 1000
1. If the company sells 1000 pens:
P = $7 . 1000 - $1000
= $6000
2. If they want to break even:
P at least = 0
Assume P = 0
=> $7.x - $1000 = 0
=> x = 1000/7 ≈ 142,86
So if they want to break even, they have to sell at least 143 pens
whatre the values of x and y
By using the concept of internal angles of a triangle, the solution of the system of linear equations related to the geometrical system is (x, y) = (40, 40).
What are the values of two variables associated with two angles from a right triangle?
In this system we have a geometrical system formed by two right triangles with a common hypotenuse. Right triangles are triangles where one of its internal angles are right angles and the sum of the measures of the internal angles within a triangle equals 180°. Then, this geometrical system can be modelled by using the following linear equations:
(x + 20) + 90 + 30 = 180 (1)
20 + 90 + (y + 30) = 180 (2)
By using the concept of internal angles of a triangle, the solution of the system of linear equations related to the geometrical system is (x, y) = (40, 40).
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What is the answer to this question?
Not sure if I'm right, but is
it RL- lol
The area of a square tin plate is 7056 square inches. Find the length of a side of the plate.
Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.
just give an clearly answers
Find f '(a).
f(x) = [tex]\frac{1}{\sqrt{x} +5}[/tex]
Answer:
[tex]f'(a)=\frac{1}{2a^{3/2}+20a+50\sqrt{a}}[/tex]
Step-by-step explanation:
[tex]f'(x)=\frac{\frac{d}{dx} (\sqrt{x}+5)(1)-(\sqrt{x}+5)\frac{d}{dx}(1)}{(\sqrt{x}+5)^2} \\ \\ =\frac{\frac{1}{2\sqrt{x}}}{(\sqrt{x}+5)^2} \\ \\ =\frac{\frac{1}{2\sqrt{x}}}{x+10\sqrt{x}+25} \\ \\ =\frac{1}{2x^{3/2}+20x+50\sqrt{x}} \\ \\ \therefore f'(a)=\frac{1}{2a^{3/2}+20a+50\sqrt{a}}[/tex]
A salesperson sold a pick-up for $18,760. They were paid a 4.5% commission. How much money were they paid?
the salesperson will be paid $ 844.2 for the one pickup they sell for $ 18,760 as 4.5 % commission is theirs.
We are given that the selling price of a pick up item is given as:
Selling price = $ 18,760
Now, it is also given that, on every sale, the salesperson gain a commission of 4.5 % as their pay.
The money they get paid for one pick up that they sell will be calculated by finding the percentage. So, we get that:
Money = 4.5 % of $ 18,760
Money = 4.5 / 100 × $ 18,760
Money = 0.045 × $ 18,760
Money = $ 844.2
Therefore, we get that, the salesperson will be paid $ 844.2 for the one pickup they sell for $ 18,760 as 4.5 % commission is theirs.
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Answer **please** ill give brainliest!
Answer:
w = -0.8
sum = 0
Step-by-step explanation:
An additive inverse is when two numbers add up to 0. 0.8+ (-0.8) = 0.
2 1/2 = 1 2/3 divided by y; what does y equal
Step-by-step explanation:
21/2=12/3
y=10.5*4
Sana nakatulong:)
A rectangular garden plot measures 12 feet by 25.6 feet.
Answer:
Do I calculate the area?
Step-by-step explanation:
This table displays the height of a balloon as it floats away. A 2-column table with 6 rows. Column 1 is labeled Time (minutes) with entries 10, 11, 12, 13, 14, 15. Column 2 is labeled Height (feet) with entries 64, 73.5, 83, 92.5, 102, 111.5. What is the rate of change for the height as the time increases by 1 minute? Rate of change = Which table shows a constant rate of change of –3? A 2-column table with 4 rows. Column 1 is labeled x with entries 3, 4, 5, 6. Column 2 is labeled y with entries negative 8, negative 5, negative 2, 1. A 2-column table with 4 rows. Column 1 is labeled x with entries 15, 16, 17, 18. Column 2 is labeled y with entries 12, 9, 7, 4. A 2-column table with 4 rows. Column 1 is labeled x with entries negative 2, negative 1, 0, 1. Column 2 is labeled y with entries negative 9, negative 12, negative 16, negative 20. A 2-column table with 4 rows. Column 1 is labeled x with entries negative 22, negative 21, negative 20, negative 19. Column 2 is labeled y with entries 2, negative 1, negative 4, negative 7.
Using the average rate of change concept, we have that:
The rate of the height is of 9.5 feet per minute.The table with a rate of -3 is given by: A 2-column table with 4 rows. Column 1 is labeled x with entries negative 22, negative 21, negative 20, negative 19. Column 2 is labeled y with entries 2, negative 1, negative 4, negative 7.What is the average rate of change of a function?The average rate of change of a function is given by the change in the output divided by the change in the input. Hence, over an interval [a,b], the rate is given as follows:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
For item 1, when the time in minutes changes by 1, the height changes by 9.5 units, hence the rate is given by:
r = 9.5/1 = 9.5 feet per minute.
For item 2, we want a table with a rate of -3, hence, when the first column changes by 1, the second column should always change by -3, hence the table is:
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 22, negative 21, negative 20, negative 19. Column 2 is labeled y with entries 2, negative 1, negative 4, negative 7.
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Researchers wanted to determine if there was an association between the level of trauma of an individual and their risk
of diabetes. The researchers studied 1544 people over the course of 12 years. During this 12-year period, they
interviewed the individuals and asked questions about their daily lives and the hassles they face. In addition,
hypothetical scenarios were presented to determine how each individual would handle the situation. These interviews
were videotaped and studied to assess the emotions of the individuals. The researchers also determined which
individuals in the study experienced any type of diabetes over the 12-year period. After their analysis, the researchers
concluded that the trauma-free individuals were less likely to experience diabetes. Complete parts (a) through (c).
(a) What type of observational study was this? Explain.
This was a
because information was collected about a group of individuals
The observational study is the Cross-sectional studies.
What is an observational study?It should be noted that an observational study is simply used to answer a research question that is based on what the researcher observes.
This is a cross-sectional studies because it analyzes a population of study at a specific point in time. It should be noted that this involves narrowing previously collected data to one point in time to test the prevalence of a theory.
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BF is dilated to created WH. The center of dilation is point A where BF=15, WH=20
What is the scale factor that was used to create WH
Answer:
The scale factor is 4/3
Step-by-step explanation:
Let x = the scale factor
15x = 20 Divide both sides by 15
x =20/15 Which simplifies to 4/3
The scale factor that was used to create WH will be 4/3.
What is the scale factor?The scale factor is defined as the proportion of the new image's size to that of the previous image decision-making. Dilation is the process of increasing the size of an object while maintaining its shape. Depending on the scale factor, the object's size can be increased or decreased.
Given that BF is dilated to create WH. The center of dilation is point A where BF=15, WH=20
The scale factor will be calculated as below:-
Let x = the scale factor
15x = 20
Divide both sides by 15
x =20/15
Which simplifies to 4/3.
x = 4 / 3
Therefore, the scale factor that was used to create WH will be 4/3.
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Show work of how $7.40 for 5 pounds
Answer:
the answer is $35.40 sir there you go
Step-by-step explanation:
that's the answer which is located above
Answer:
1.48 each
Step-by-step explanation:
you have to divide 7.40/5
Kitchen
Living
Room
Bathroom
Laundry
Bedroom #2
Bedroom #1
Scale: 2 units = 3 feet
to much long and I want brainliest
A label for a drug indicates that each milliliter contains 7 milligrams of the drug. How many milligrams are in 8 milliliters?
There are 56 milligrams in 8 milliliters of the drug if the drug's label says each milliliter contains 7 milligrams of the substance.
What is Ratio?The ratio is defined as a relationship between two quantities, it is expressed as one divided by the other.
Let x milligrams would in 8 milliliters.
We can write this as a ratio as
As per the given information,
1 milliliter : 7 milligrams = 8 milliliter : x milligrams
⇒ 1 / 7 = 8 / x
⇒ x = 8 × 7
⇒ x = 56 milligrams
Hence, there are 56 milligrams in 8 milliliters of the drug.
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Stop light A changes color every 45 seconds and Stop light B changes color every 80 seconds. Both Stop lights just changed. After how many seconds will the stoplights change color at the same time?
The stoplights will change color after 720 seconds. at the same time again.
Given: Stoplights A and B alternate between changing colors every 45 and 80 seconds, respectively.
One copy of each of the unique components and the common factor make up the Least Common Multiple (LCM).
The least common multiple of 45 and 80 is the amount of seconds before they change color simultaneously once more.
Since 80 = 5 x 2 x 2 x 2 and 45 = 5 x 3 x 3 respectively
A least common multiple of 45 and 80 is equal to 2x2x2x2x3x3x5 = 720.
Hence, the color of the spotlight will change after 720 seconds. at the same time again.
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Please help i really need this done :) I added 50 p[oints
A) Matching the stages of a business life cycle in the correct order is as follows:
ExistenceSurvivalSuccessTake-off Pivot or persistResource MaturityExit Plan.B) Matching the business life cycle to its description is as follows:
Life Cycle DescriptionExit Plan The owner decides to sell.
Success A business owner gets it running and introduces it
to the market in its attempt to catch the competition.
What is the business life cycle?The business life cycle refers to the gradual progress of a business over its lifetime.
A business life cycle can be divided into five broad stages: launch, growth, shake-out, maturity, and decline. Some management gurus classify the life cycle of a business into the following stages: Existence, Survival, Success, Take-off, and Resource Maturity.
Thus, a successful business declines after its maturity stage.
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