find the critical points of f(x, y) = 2x 2 y 2 − x 2 y − 3y and analyze them using the second derivative test.
Answers
The critical points of f(x, y) = 2x 2 y 2 − x 2 y − 3y and analyze them using the second derivative test (0, 3/4) is a local maximum, and (1/2, 1/2) is a local minimum.
To find the critical points, we need to solve the system of equations:
∂f/∂x = 4xy² - 2xy = 2xy(2y-1) = 0
∂f/∂y = 4x²y - x² - 3 = x²(4y-1) - 3 = 0
The critical points are (0, 0), (0, 3/4), and (1/2, 1/2).
To analyze these points using the second derivative test, we need to compute the second partial derivatives:
f_xx = 4y² - 2y
f_xy = f_yx = 4xy - x
f_yy = 4x²
At (0, 0), we have f_xx(0, 0) = 0, f_xy(0, 0) = 0, and f_yy(0, 0) = 0. The second derivative test is inconclusive.
At (0, 3/4), we have f_xx(0, 3/4) = -3/2, f_xy(0, 3/4) = 0, and f_yy(0, 3/4) = 0. The Hessian matrix is negative definite, which means that (0, 3/4) is a local maximum.
At (1/2, 1/2), we have f_xx(1/2, 1/2) = 1, f_xy(1/2, 1/2) = 0, and f_yy(1/2, 1/2) = 1. The Hessian matrix is positive definite, which means that (1/2, 1/2) is a local minimum.
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Related Questions
Select the correct answer.
Which expression is equivalent to
3a2-6a/3a3 ? Assume that the denominator does not equal zero
Answers
the final equivalent expression is 2(a - 1) / 3a2 for the expression 3a2-6a/3a3 can be simplified by factoring out the greatest common factor in the numerator and denominator.
The greatest common factor of the numerator is 3a, which can be factored out as follows:
3a(2a - 2) / 3a(3a2)
The 3a in the numerator and denominator cancel out, leaving:
2a - 2 / 3a2
Therefore, the expression is equivalent to (2a - 2) / (3a2).
This expression can be further simplified by factoring out a 2 in the numerator:
2(a - 1) / 3a2
So, the final equivalent expression is 2(a - 1) / 3a2.
In summary, the expression 3a2-6a/3a3 simplifies to 2(a - 1) / 3a2, which is the same as 2a - 2 / 3a2 after factoring out the greatest common factor. It is important to note that we assume the denominator does not equal zero because division by zero is undefined and not a valid mathematical operation.
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If G( x ) = 3 x + 1, then G -1 (1) is...
-1/3
-4
-0
Answers
Answer:
-4
Step-by-step explanation:
3+1 = 4 so just put a dash at the start and you ger -4
How should the values below be reported? 25.09874±0.0793
Answers
The value should be reported as follows:
25.1 ± 0.1
In this case, the value has been rounded to the same decimal place as the uncertainty, which is the first decimal place. The uncertainty is rounded to one significant figure to reflect the precision of the measurement. The value is reported with one decimal place, which is the same as the uncertainty, to maintain consistency.
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By coincidence, Kimberly’s Paint Supplies sold equal quantities of two paint colors yesterday: yellow and blue. All yellow paint comes in 9-liter containers while blue paint is sold in 12-liter containers. What is the smallest amount of each paint color the store must have sold?
Answers
Answer:
36 liters
Step-by-step explanation:
Yellow paint = 9 liters containers
Blue paint = 12 liters containers
Kimberly’s Paint Supplies sold equal quantities of two paint colors yesterday
What is the smallest amount of each paint color the store must have sold?
Find the Lowest common multiple of 9 liters and 12 liters
9 liters = 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
12 liters = 24, 36, 48, 60, 72, 84, 96, 108
The Lowest common multiple of 9 liters and 12 liters is 36 liters
Therefore, the smallest amount of each paint color the store must have sold to have equal quantities of two paint colors yesterday is 36 liters
Answer: The answer is 9.
Step-by-step explanation:
Because I got it right on ixl.
Consider the function f(x) = 4x^3 – 3x on the interval [ -5,5). Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists at least one c in the open interval (-5,5) such that f'(c) is equal to this mean slope. For this problem, there are two values of C that work. The smaller one is and the larger one is
And the larger one is
Answers
There are two values of c that work: the smaller one is c = -√(103 / 12), and the larger one is c = √(103 / 12).
To find the average or mean slope of the function f(x) = 4\(x^3\) - 3x on the interval [-5,5), we need to follow these steps:
1. Calculate the difference in the function's value between the endpoints.
2. Divide this difference by the width of the interval.
3. Use the Mean Value Theorem to find the values of c where f'(c) is equal to the mean slope.
Step 1: Calculate the difference in the function's value between the endpoints.
f(5) = \(4(5)^3\) - 3(5) = 500
f(-5) = \(4(5)^3\) - 3(-5) = -500
Difference: f(5) - f(-5) = 500 - (-500) = 1000
Step 2: Divide the difference by the width of the interval.
Width of the interval = 5 - (-5) = 10
Mean slope = Difference / Width = 1000 / 10 = 100
Step 3: Use the Mean Value Theorem to find the values of c where f'(c) is equal to the mean slope.
First, find the derivative of f(x): f'(x) = \(12x^2\) - 3
Now, set f'(c) equal to the mean slope and solve for c:
100 = \(12c^2\) - 3
103 = \(12c^2\)
\(c^2\) = 103 / 12
c = ±√(103 / 12)
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Parabola A can be represented using the equation (x + 3)2 = y, while line B can be represented using the equation y = mx + 9. Isabel claims one solution to the system of two equations must always be the vertex of parabola A. Which best describes the reasonableness of her claim?
Answers
The answer is not A!
Answer: The correct answer is c, Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex.
consider a 3x3 matrix a this matrix has -2 as an eigen value compute a basis of eigen space corresponding to eigen value -2
Answers
To compute a basis of the eigen space corresponding to eigen value -2, we need to find the null space of the matrix A + 2I, where A is the 3x3 matrix and I is the identity matrix.
The null space will give us the basis vectors of the eigen space
To find the eigen space corresponding to the eigen value -2, we start by constructing the matrix A + 2I, where A is the given 3x3 matrix and I is the 3x3 identity matrix. Next, we solve the homogeneous system of linear equations (A + 2I)x = 0, where x is a vector. The solutions to this system form the null space of the matrix A + 2I.
By finding a basis for this null space, we can obtain the basis vectors of the eigen space corresponding to the eigen value -2.
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Asolve this system of equations using a matrix inverse. if a is the coefficient matrix of the above system of equations and , then the elements of are , the elements of are , and the elements are . the solution of this system is (x, y, z) = .
Answers
The solution of this system is represented by the values of (x, y, z).
To solve the system of equations using a matrix inverse, we first need to find the inverse of the coefficient matrix, denoted as "A". Let's assume that "A" is a 3x3 matrix.
Find the determinant of matrix A:
det(A) = ad - bc
Check if the determinant is non-zero. If det(A) ≠ 0, then A has an inverse.
Calculate the matrix of minors:
M = |d -b a |
|-f e -c|
|i -h g |
Calculate the matrix of cofactors:
C = |d -b a |
|-f e -c|
|i -h g |
Calculate the adjugate matrix:
Adj(A) = C^T (transpose of C)
Find the inverse of A:
A^-1 = (1/det(A)) * Adj(A)
Multiply the inverse matrix by the constant matrix:
X = A^-1 * B
The solution of this system is represented by the values of (x, y, z).
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a bottle manufacturer has determined that the cost (c) in dollars of producing x bottles is c=0.55x+2100 . what is the cost of producing 400 bottles?
Answers
The cost of producing 400 bottles is $2320 .
What is cost function ?
The cost that will be incurred at a specific activity level is predicted using a cost function. This formula usually only works within a certain range of activity levels; over that point, it stops producing reliable results. Beyond these activity levels' outer limits, the cost function must be changed to take into account things like changes in volume discounts and the occurrence of step charges.
Here the given function shows the cost in dollars for x number of bottles.
=> c = 0.55x + 2100
Here number of bottles x= 400
=> c = 0.55* 400 + 2100 = $2320
Hence the cost of producing 400 bottles is $2320.
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Using the 45°-45°-90° triangle theorem, find the value of h, the height of the wall. 6.5 ft 6.5 startroot 2 endroot ft 13 ft 13 startroot 2 endroot ft
Answers
The height of the wall is 13 ft if the triangle is 45°-45°-90° option third 13 ft is correct.
What is a right-angle triangle?It is a triangle in which one of the angles is 90 degrees and the other two are sharp angles. The sides of a right-angled triangle are known as the hypotenuse, perpendicular, and base.
We have a 45°-45°-90° triangle.
As we know, in a triangle with angles of 45°- 45° - 90° the sides of that triangle are in the proportion x : x : x√2.
From the diagram and ratio:
h√2 = 13√2
h = 13 ft
Thus, the height of the wall is 13 ft if the triangle is 45°-45°-90° option third 13 ft is correct.
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Answer:C
Step-by-step explanation:
edge
Simplify −5(2x−6)+25x . Write your answer in factored form.
Answers
Answer:
-5(2x-6)+25x
-10x+30+25x
25x-10x+30
15x+30
15(x+2)
Step-by-step explanation:
The factored form of the given expression is 15(x + 2).
What is an Algebraic expression?An algebraic expression can be obtained by doing mathematical operations on the variable and constant terms.
The variable part of an algebraic expression can never be added or subtracted from the constant part.
The given expression is as below,
−5(2x − 6) + 25x
It can be simplified as follows,
−5(2x − 6) + 25x
= -10x + 30 + 25x
= 15x + 30
= 15(x + 2)
Hence, the given expression can be simplified in factored form as 15(x + 2).
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Simplify (-64)2/3.
Answers
Answer:
As a proper fraction. (numerator smaller than denominator): 2/3 = 2/3
As a decimal number: 2/3 ≈ 0.67.
As a percentage: 2/3 ≈ 66.67%
That's all I can provide
The value of the expression (-64)^(2/3) can be simplified as 16.
Given an expression (-64)^(2/3).
We have to find the value of the expression.
We have the rules of exponents that,
aᵇⁿ = (aᵇ)ⁿ
Using this rule, the expression can be written as,
(-64)^(2/3) = (-64²)^(1/3)
= (4096)^(1/3)
Now,
(a)^(1/3) is equal to ∛a.
(-64)^(2/3) = ∛4096
= 16
Hence the correct option is B.
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Simplify the following expression: 3x+3(x+6)−4−5x Responses
Answers
Answer:
3x + 3x + 18 - 4 - 5x
x + 18 - 4
x + 14
x + 14 is the answer
Will Mark Brainliest!!! Name the property illustrated.
Answers
Answer:
5) distributive property
7) Transitive property
8) Associative property
Step-by-step explanation:
if i was born march 4 was horoscope am i?
and please give a description also
Answers
Answer:
Pisces
Step-by-step explanation:
you are well known for your intuitive, self-sacrificing and sensitive personality. At times you may feel psychic, but this is only because of your innate ability to sense the moods and thoughts of others.
Answer:
Step-by-step explanation:
March 4 Zodiac is Pisces - Full Horoscope Personality Being a Pisces born on March 4th, you believe in hidden meanings and hope your behaviour is going to help you in life by gathering good karma around you. At times, your behaviour is dominant but generally you are a romantic and a genuinely loyal person.
let y = 2e^cosx both x and y vary with time in such a way that y increases at the constant rate of 5 units per secobnd. the rate at which x is changing when x = pi/2
Answers
When x = π/2, the rate at which x is changing can be calculated by using the chain rule. The rate at which x is changing is equal to \(-5e^{(-sin(\pi /2))\), or -5.
We are given that \(y = 2e^{cos(x)\) and that y is increasing at a constant rate of 5 units per second. To find the rate at which x is changing when x = π/2, we need to differentiate y with respect to time using the chain rule.
Using the chain rule, we differentiate \(y = 2e^{cos(x)\) as follows: dy/dt = dy/dx * dx/dt. Since we know that dy/dt is 5 units per second, we can rewrite the equation as 5 = dy/dx * dx/dt.
To find dx/dt when x = π/2, we substitute x = π/2 into the equation. Now we need to find dy/dx. Taking the derivative of \(y = 2e^{cos(x)\) with respect to x, we get \(dy/dx = -2e^{cos(\pi /2)} sin(\pi /2)\)
Substituting x = π/2 into dy/dx, we have \(dy/dx = -2e^{cos(\pi /2)} sin(\pi /2)\). Since cos(π/2) = 0 and sin(π/2) = 1, we can simplify dy/dx to -2e⁰ * 1 = -2.
Finally, we can rearrange the equation 5 = dy/dx * dx/dt and substitute dy/dx = -2 to solve for dx/dt. We get -2 * dx/dt = 5, which implies dx/dt = -5/2 or -2.5.
Therefore, when x = π/2, the rate at which x is changing is -2.5.
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can someone please help me find the domain and range
Answers
Answer:
the domains are x axis or intercepts and range is y axis or intercepts
Identify the domain and the range of the relation shown in the figure
Answers
Step-by-step explanation:
Where is figure.
Plz attached pictures.
Here is a scatter plot: The graph of what linear equation is a good fit for this data?
Answers
Answer:
D
Step-by-step explanation:
Y=1/2x because the line is going straight up diagnolly and the 1 makes it higher
14. A random sample of 25 seventh graders found
that 10 enjoy participating in a local contest
making duct-tape wallets.
a. If there are 480 seventh graders, write a
proportion to estimate the number of seventh
graders, n, who participate in the contest.
b. Solve the proportion to estimate the number
of seventh graders who participate in the
contest.
c. Write and solve a proportion to estimate the
number of seventh graders, n, who do not
participate in the contest.
Answers
a. The proportion to estimate the number of seventh graders who participate in the contest is $\frac{10}{25} = \frac{n}{480}$.
b. To solve the proportion, we can multiply both sides by 480 to get $n = \frac{10 \cdot 480}{25} = 192$.
c. The proportion to estimate the number of seventh graders who do not participate in the contest is $\frac{15}{25} = \frac{n}{480}$. To solve this proportion, we can multiply both sides by 480 to get $n = \frac{15 \cdot 480}{25} = 288$.
state the law of logic represented: if you exercise every day, then you will be a better athlete. You exercise every day. So, you will be a better athlete. a. law of syllogism b. law of detachment c. neither
Answers
Answer:
Step-by-step explanation:
c
Choose if each statement is True or False. {(2, 2), (3, 2), (4, 2), (6, 2)} is a function: {(-1, 5), (0, 8), (3, 12), (6, 21)} is a function: 5 -10 0 -5 15 -10- 5 0 -5 -10+ 5 5 10 is a function: 10 15 is a function:
Answers
Answer:
True False True False
Step-by-step explanation:
who is the best rapper
Answers
Answer:
Eminem.
Rakim.
Nas.
Andre 3000.
Lauryn Hill.
Ghostface Killah.
Kendrick Lamar.
Lil Wayne
Step-by-step explanation:
Answer:
Logic
Step-by-step explanation:
tell whether the ratios form a proportion 4:14 and 12:40
Answers
Answer:
NO
Step-by-step explanation:
First, we should simplify both fractions to the most simple form.
4/14=2/7
12/40=6/20=3/10
A proportion means that it is equal. 2/7 does not equal 3/10, so the answer is no.
Solve: 2x + 6 + 10 = 12
Answers
Answer:
x= -2
Step-by-step explanation:
2x + 6 + 10=12
2x + 16=12
2x=-4
x=-2
Chase is moving and must rent a truck. There is an initial charge of $35 for the rental plus a fee of $2.50 per mile driven. Make a table of values and then write an equation for C,C, in terms of m,m, representing the total cost of renting the truck if Chase were to drive m miles.
Answers
The required equation in the given situation is C = 35 + 2.50m where C is the total cost and m is the number of miles driven.
What is the equation?Equation: A declaration that two expressions with variables or integers are equal.
In essence, equations are questions and attempts to systematically identify the solutions to these questions have been the driving forces behind the creation of mathematics.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign.
A formula would be 3x - 5 = 16, for instance.
The equation would be:
C is the total cost and m is the miles driven.
We know that:
Charge of the truck: $35
Charge per mile: $2.50
Then, form the equation as follows:
C = 35 + 2.50m
Therefore, the required equation in the given situation is C = 35 + 2.50m where C is the total cost and m is the number of miles driven.
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Today only, a sofa is being sold at a 27% discount. The sale price is $511. What was the price yesterday?
Answers
Since we have that the sale price is $511, then this represents the 100-27%=73% of the total price. Then, using a rule of three we have:
\(\begin{gathered} 511\rightarrow73\text{ \%} \\ x\rightarrow100\text{ \%} \\ \Rightarrow x=\frac{100\cdot511}{73}=\frac{51100}{73}=700 \\ x=700 \end{gathered}\)therefore, the original price of the sofa was $700
Rafael needs to drive 16 miles to work.so far, he has driven 4.1 miles.How many more miles must he drive
?
Answers
Answer:
11.9 miles
Step-by-step explanation:
16 - 4.1 = 11.9
What is the y-intercept of the line with the equation below?
Y = 10x - 7
Answers
Answer:
-7
Step-by-step explanation:
the y intercept is the value that is not a foefficient of x or y
answer is: -7
Alex is correct
The left and right ends of the normal probability distribution extend indefinitely, never quite touching the horizontal axis. True False
Answers
It is false as the left and right ends of the normal probability distribution extend indefinitely, approaching but never touching the horizontal axis.
The statement is false because the left and right ends of the normal probability distribution do not extend indefinitely. In reality, the normal distribution is defined over the entire real number line, meaning it extends infinitely in both the positive and negative directions. However, as the values move further away from the mean (the center of the distribution), the probability density decreases. This means that although the distribution approaches but never touches the horizontal axis at its tails, the probability of observing values extremely far away from the mean becomes extremely low. Thus, while the distribution theoretically extends infinitely, the practical probability of observing values far from the mean decreases rapidly.
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