1. Minimise the function f(x,y) = x2 + 2y? subject to x + y 2, X Sy using the Karush-Kuhn-Tucker conditions.
Answers
Therefore, the optimal solution is (x,y) = (1,1), with a minimum value of f(1,1) = 3.
Given the function f(x,y) = x^2 + 2y;
subject to x + y <= 2, x, y >=0
The KKT conditions are as follows:
1. g(x,y) = x + y - 2 <= 0 and h1(x,y)
= -x <= 0, h2(x,y)
= -y <= 0 and
f(x,y) = x^2 + 2y2.
λ1 >= 0,
λ2 >= 0 and
μ1 >= 0,
μ2 >= 0.3.
λ1g(x,y) = λ1(x + y - 2)
= 0;
h1(x,y) = -x
= 0;
h2(x,y) = -y
= 0.4.
∇f(x,y) - λ1∇g(x,y) - μ1∇h1(x,y) - μ2∇h2(x,y) = 0; at optimal x,y.
Here is a detailed solution:
Solution:
Let's begin by stating the Lagrangian function:
L(x,y, λ1, λ2, μ1, μ2) = x^2 + 2y + λ1(2 - x - y) - μ1x - μ2y
First, we need to find the partial derivatives:
∂L/∂x = 2x - λ1 - μ1
∂L/∂y = 2 - λ1 - μ2
∂L/∂λ1 = 2 - x - y
∂L/∂λ2 = 0
∂L/∂μ1 = -x
∂L/∂μ2 = -y
Now, we set the derivatives to zero:
2x - λ1 - μ1 = 0
2 - λ1 - μ2 = 0
2 - x - y = 0
x + y - 2 <= 0
-x <= 0
-y <= 0
Next, we use the KKT conditions and find λ1, λ2, μ1, and μ2:
λ1(x + y - 2) = 0
-μ1x = 0
-μ2y = 0
If λ1 = 0, then we must have x + y - 2 <= 0, which means that x + y = 2.
If λ1 > 0, then we must have x + y - 2 = 0, which means that x + y = 2. From (1) and (2), we get that λ1 = 2 and μ1 = 0.
If λ2 > 0, then μ2 = 0, which means that ∂L/∂y = 0. Substituting λ1 = 2 and μ2 = 0 into (2), we get that y = 1.
Then, from (1), we get that x = 1. Finally, μ1 and μ2 are zero, so the KKT conditions are satisfied.
To know more about Karush-Kuhn-Tucker visit:
https://brainly.com/question/32542902
#SPJ11
Related Questions
HELP ASAP 61/35 as a mixed number
Answers
Answer:
1 26/35
Step-by-step explanation:
61/35 =
1 26/35
Hope that helps!
Answer:
1 26/35
Step-by-step explanation:
So there is 1 whole number and 26/ 35
i don't know if that helps but thats helps but that's the answer
Where in a report of a quantitative research study would a nurse expect to find the gaps or conflicts about the phenomenon studied to be identified?
Answers
In a report of a quantitative research study, a nurse would expect to find the gaps or conflicts about the phenomenon being studied to be identified in: D. Review of the literature.
What is a research report?A research report can be defined as a publication that is used by a researcher to provide data about a research work after analyzing the information gathered on a particular subject, especially in a summarized or condensed format..
The types of research report.In Science, there are three main types of research report and these include the following:
Presented research report.Refereed research report.Reviewed research report.What is a review of the literature?A review of the literature is also referred to as literature review and it is typically used by a researcher to describe what's known and what isn't known about a phenomenon that is being studied.
Read more on research report here: https://brainly.com/question/26177190
#SPJ1
Complete Question:
Where in a report of a quantitative research study would a nurse researcher expect the gaps or conflicts about the phenomenon studied to be identified?
a. Analysis of data
b. Research design
c. Problem statement
d. Review of the literature
how do yo do this question I have to
Solve the System by Substitution:
y = x-2
x + y = 4
Answers
14. In exercise 13, suppose grapes cost
$2.35 per pound, oranges cost $0.99 per
pound, and apples cost $1.65 per pound.
Estimate how much Adrian paid for all
the fruit.
Answers
Assume grapes cost $2.35 per pound, oranges cost $0.99 per pound, and apples cost $1.65 per pound in exercise 13. Adrian paid a total of $5 for the fruits.
What is Estimation?Calculations are made easier and more realistically when a number is estimated, which is a plausible estimation of its true value.To estimate something is to approximate it as accurately as possible.To acquire a quick and approximate response, this is obtained by rounding off the values used in the calculation. Rounding integers to the closest whole number is the simplest method for estimating. Rounding off makes getting a rough answer quicker. Koel, for instance, makes a 3950 monthly deposit toward the cost of her new home. This value can be rounded out to 4000 to make the calculation easier. To round a number, determine its nearest required digit. To avoid working with fractions or decimals, numbers are frequently rounded to the next whole number.Therefore,
The cost of Grapes = $2.35 per pound
The cost of Oranges = $0.99 per pound
The cost of Apples = $1.65 per pound
If Adrian purchased one pound of each fruit,
He paid:
$2.35 + $0.99 + $1.65 = $4.99 = $5 (Approx.)
So, Adrian paid $5 in total for the fruits.
To learn more about Estimation, refer to:
https://brainly.com/question/15338244
#SPJ13
Which rational number equals 0.1?
1/11
1/10
1/9
1/8
HELPPPPPP PLZZZZZ
Answers
All you have to do is divide the numerator by the denominator
In this case 1 divided by 10 = 0.1
Hope this helps
If a 3 by 3 matrix has det A = -1, find det(\frac{1}{2}A) and det(-A) and det({A}^{2}) and det({A}^{-1} ).
Answers
If a 3 by 3 matrix has det A = -1, then det (1/2 A) = -1/8, det (-A) = -det (A) = 1, det (A²) = det (AA) = det (A) × det (A) = (-1) × (-1) = 1, and det (A⁻¹) = 1/det (A) = -1.
These results can be shown as follows:
Given that det A = -1, the matrix A is invertible, meaning that A has an inverse, A⁻¹. We can use this fact to find the determinants of the matrices 1/2 A and -A.
To find the determinant of 1/2 A, we use the fact that det (kA) = k³ det A for any scalar k and any matrix A. Thus, det (1/2 A) = (1/2)³ det A = (-1/8)
To find the determinant of -A, we use the fact that det (-A) = (-1)³ det A = -det A = -(-1) = 1
To find the determinant of A², we use the fact that det (AB) = det A × det B for any matrices A and B of the same size. Thus, det (A²) = det (AA) = det A × det A = (-1) × (-1) = 1
To find the determinant of A⁻¹, we use the fact that det (A⁻¹ A) = det I = 1.
Thus, det A⁻¹ × det A = 1, which implies that det A⁻¹ = 1/det A = -1
To know more about matrix visit:
https://brainly.com/question/29132693
#SPJ11
The value of a bank account is −$28. Money is then deposited into the account. Which could represent the value of the account now?
Answers
Answer:
the money could be positive now, such like you added money to the negative to make it highly ig
a study is to be conducted to help determine whether a die is fair. how many degrees of freedom are there for a chi-square goodness-of-fit test?
Answers
The degrees of freedom for a chi-square goodness-of-fit test are calculated as the number of categories minus 1.
Suppose we wish to determine if an ordinary-looking six-sided die is fair, or balanced, meaning that each face has a probability of 1/6
of arrival on top when the die is tossed. We could toss the die dozens, maybe hundreds, of times and compare the actual number of times each face arrival on top to the scheduled number, which would be 1/6
of the total number of tosses. We would not expect each number to be exactly 1/6 of the total, but it should be close.
To know more about Chi-Square test:
https://brainly.com/question/14082240
#SPJ4
Describe the following transformation. Select all that apply
g(x) = -2(x + 5)
Answers
The transformation g(x) = -2(x + 5) is described as follows
vertical shift up 5 unitshorizontal shift left 5 unitsreflection over the y axisvertical stretch by a factor of 2What is transformation?Transformation is a term used in mathematics to refer to the movements made by objects.
Translation refers to a type of transformation that involve a linear movement.
The following movements are involved in translation
leftrightupdownhe left and right movements are on the horizontal direction controlled by the x coordinate. (x + 5) = horizontal shift left 5 units
While the up and down movements are on the vertical direction controlled by the y coordinate. k(x + 5) vertical shift up 5 units
multiplying the equation by a number greater than one leads to vertical stretch since it is a negative number this will be a reflection over the y axis
Learn more on translation:
https://brainly.com/question/29321966
#SPJ1
state the degree of f(x)= -x^4+x^2+x+1
Answers
Answer:
degree of 4
Step-by-step explanation:
The largest exponent.
a tank in the shape of a hemisphere has a diameter of 6 feet. if the liquid that fills the tank has a density of 53.3 pounds per cubic foot, what is the total weight of the liquid in the tank, to the nearest full pound?
Answers
The total weight of the liquid in the tank, to the nearest full pound, is calculated to be 3,012 pounds.
The volume of a hemisphere with diameter 6 feet is (2/3) x π x (6/2)³ = 56.55 cubic feet.
The weight of the liquid in the tank can be found by multiplying the volume of the liquid by its density:
Weight = Volume x Density = 56.55 x 53.3 = 3,012.32 pounds.
Rounding this to the nearest full pound gives a total weight of 3,012 pounds. Therefore, the total weight of the liquid in the tank is 3,012 pounds (to the nearest full pound).
Learn more about Hemisphere :
https://brainly.com/question/30480557
#SPJ4
Answer:
3014
Step-by-step explanation:
I just did it
What is multiplicative property of equality?
Answers
According to this property, when both sides of an equation are multiplied by the same real number, both sides of the equation always remain the same. The formula for this property can be expressed in real numbers a, b and c. If a × c = b × c.
Property of Equality:
The equivalence property describes the relationship between two equal quantities. If you apply a math operation to one side of the equation, you must also apply it to the other side of the equation to maintain balance.
That is, a property that does not change the truth value of an equation or does not affect the equivalence of two or more quantities is called an equality property. These equality properties help solve various algebraic equations and define equivalence or equilibrium relationships.
Multiplicative Property of Equality:
According to this property, when both sides of an equation are multiplied by the same real number, both sides of the equation always remain the same.
The formula for this property can be expressed as for the real numbers a, b, and c.
If a × b, then, a × c = b × c.
In algebra, the multiplicative property of an equation helps extract unknown terms from an equation. Because multiplication and division are opposites of each other.
Learn more about real number:
https://brainly.com/question/10547079
#SPJ4
X=______ ft heIp pIz
Answers
Answer: x= 67.5 feet
Step-by-step explanation:
Write 60.9% as a decimal.
A. 0.0609
B. 0.609
C. 0.00609
D. 6.0900
Answers
Answer:
0.609
Step-by-step explanation:
convert to a fraction then convert to a decimal by dividing the numerator and denominator :)
Explaination: By definition p%= p/100
60.9/100
Convert the fraction into a decimal
0.609
A bag contains 5 red, 5 blue, 4 green, and 2 yellow marbles. What are the odds of drawing a red marble? *
1 point
1:1
5:11
1:3
5:9
Answers
Answer:
5:11
Step-by-step explanation:
you would say 5 marbles to the amount of the other marbles you have which would be 11
What value of x will make the number 40 35 :: 16 x?
Answers
Answer:
Since 40:35 simplifies to 8:7, 16:X = 8:7, implies X = 14. Hope this helps you!
Step-by-step explanation:
hope this helps
mark me brainliest
A copy machine makes 36 copies per minute. How many copies does it make in 5 minutes and 15 seconds? Clears your work. Undoes your last action.
Answers
To solve this problem, we need to first convert 5 minutes and 15 seconds to a decimal form of minutes.
5 minutes is equal to 5.0 minutes, and 15 seconds is equal to 15/60 = 0.25 minutes. Therefore, the total time in minutes is:
5.0 + 0.25 = 5.25 minutes
Next, we can multiply the number of copies made per minute by the total time in minutes to find the total number of copies made:
36 copies/minute x 5.25 minutes = 189 copies
Therefore, the copy machine makes 189 copies in 5 minutes and 15 seconds.
What is the coefficient of 5x² in 5x² 5x?
Answers
Answer:
Step-by-step explanation:
i think 1
Noise levels at 5 airports were measured in decibels yielding the following data:
162,176,162,141,159
Construct the 90% confidence interval for the mean noise level at such locations. Assume the population is approximately normal.
Step 1 of 4:
Calculate the sample mean for the given sample data. Round your answer to one decimal place.
Step 2 of 4:
Calculate the sample standard deviation for the given sample data. Round your answer to one decimal place.
Step 3 of 4:
Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.
Step 4 of 4:
Construct the 90% confidence interval. Round your answer to one decimal place.
Answers
The 90% confidence interval for the mean noise level at these airport locations is approximately (148.1, 171.9).
Step 1 of 4:
To calculate the sample mean for the given sample data, we sum all the values and divide by the sample size.
Sample mean = (162 + 176 + 162 + 141 + 159) / 5 ≈ 160.0 (rounded to one decimal place)
Step 2 of 4:
To calculate the sample standard deviation, we need to find the differences between each data point and the sample mean, square those differences, sum them up, divide by the sample size minus 1, and then take the square root.
Deviation from mean: (162 - 160)^2, (176 - 160)^2, (162 - 160)^2, (141 - 160)^2, (159 - 160)^2
Sum of squared deviations = (2^2 + 16^2 + 2^2 + (-19)^2 + (-1)^2) = 590
Sample standard deviation = sqrt(590 / (5 - 1)) ≈ 9.6 (rounded to one decimal place)
Step 3 of 4:
To find the critical value for a 90% confidence interval, we need to look up the t-value in the t-distribution table with (n-1) degrees of freedom. Here, n is the sample size, which is 5.
Degrees of freedom = 5 - 1 = 4
Looking up the t-value for a 90% confidence level and 4 degrees of freedom, we find it to be approximately 2.776 (rounded to three decimal places).
Step 4 of 4:
To construct the 90% confidence interval, we use the formula:
Confidence interval = sample mean ± (critical value * (sample standard deviation / sqrt(sample size)))
Confidence interval = 160.0 ± (2.776 * (9.6 / sqrt(5)))
Calculating this expression will give us the lower and upper bounds of the confidence interval.
Confidence interval = 160.0 ± (2.776 * 4.297) ≈ 160.0 ± 11.9 (rounded to one decimal place)
Using the sample mean, standard deviation, and critical value, we constructed a 90% confidence interval for the mean noise level, which is approximately (148.1, 171.9) decibels. This confidence interval provides an estimate of the range within which the true mean noise level at these airports is likely to fall with 90% confidence.
To know more about confidence interval refer here:
https://brainly.com/question/32278466?#
#SPJ11
Jordan saved $225 and continues to save $45 each week. RayAnn saved $150 and continues to save $60 each week. Write and solve a system of equations using any method for the given situation.
Equation for Jordan:
Equation for RayAnn:
Solution:
??please help???
Answers
The equations using any method for the given situation illustrated that:
The equation for Jordan: 225 + 45w
The equation for RayAnn: 150 + 60w
How to calculate the equation?Let the number of weeks be represented by w. Jordan saved $225 and continues to save $45 each week. The equation for this will be:
= 225 + (45 × w)
= 225 + 45w
RayAnn saved $150 and continues to save $60 each week. The equation will be:
= 150 + (60 × w)
= 150 + 60w
Learn more about equations on:
brainly.com/question/2972832
#SPJ1
Bob bought a computer in California the computer cost $600 California has an 8% sale tax how much did bob pay for the computer including sale tax
Answers
Answer:
Below
Step-by-step explanation:
600 = cost
.08 * 600 = tax
Add the two together 600 + .08 (600) = 1.08 * 600 = $ 648
Salmonella bacteria grow rapidly at room temperature. After a contaminated chicken is cut up, a population of bacteria are left on a cutting board ending up in a salad. Suppose that the number present in the salad after t hours is given by S(t)= 250 x 6t How often do the bacteria double? State the answer in hours using four significant figures.
Answers
As per the temperature, the population of Salmonella bacteria in the salad will double approximately every 2.87 hours if the conditions for growth remain constant.
The population of Salmonella bacteria can increase over time if they have the necessary nutrients and temperature for growth. In this case, we have a cutting board contaminated with Salmonella bacteria, which ends up in a salad. The number of bacteria present in the salad after t hours is given by the formula S(t) = 250 x 6t, where t is the time in hours.
To understand how often the population of bacteria doubles, we need to look at the growth rate of the bacteria.
To calculate the doubling time, we can use the formula Td = ln(2) / r, where Td is the doubling time, ln is the natural logarithm, and r is the exponential growth rate. In this case, the growth rate can be found by taking the derivative of S(t) with respect to t, which gives us r = 6ln(6).
Plugging in the values, we get Td = ln(2) / 6ln(6) = 2.87 hours
To know more about temperature here
https://brainly.com/question/11464844
#SPJ4
C-Ice is a canned beverage made from tea and cannabis that is marketed in Europe as a nutritional drink that can boost the user's immune system. It can only be purchased at health food stores. This limitation on the _____ element of its marketing mix strategy supports the product’s competitive advantage.
a. planning
b. product
c. promotion
d. distribution
e. production
2. Which of the following is NOT an example of a product's tangible feature?
a. brand equity
b. packaging
c. color
d. weight
e. size
Answers
This limitation on the distribution element of its marketing mix strategy supports the product’s competitive advantage.
The one which is not representing an example of a product's tangible feature is brand equity.
The limitation on the distribution element of the marketing mix strategy supports the product's competitive advantage.
By exclusively selling C-Ice at health food stores,
The company creates a selective distribution channel that positions the product as a specialized and premium offering.
This limited availability enhances the perception of exclusivity.
And uniqueness, potentially attracting health-conscious consumers who value natural and nutritional products.
Brand equity is not an example of a product's tangible feature.
Brand equity refers to the intangible value and perception associated with a brand,
Such as its reputation, customer loyalty, and brand recognition.
Tangible features, on the other hand, are physical characteristics of a product that can be observed or measured.
Examples of tangible features include packaging, color, weight, and size.
Learn more about marketing here
brainly.com/question/31311464
#SPJ4
A noticeably different form or state of the same substance.
a. kinetic energy
b. molecule
c. temperature
d. phase
e. freedom of movement
Answers
A noticeably different form or state of the same substance is called phase.
A phase refers to a noticeably different form or state of the same substance. For example, water can exist in three different phases: solid (ice), liquid (water), and gas (water vapor). Each phase has distinct properties and characteristics, such as different arrangements of molecules and different levels of freedom of movement. Water can exist in three primary phases: solid, liquid, and gas.
Solid Phase (Ice): At low temperatures, water molecules slow down and come together in a regular, ordered arrangement. This results in the formation of a solid phase known as ice. In the solid phase, water molecules are closely packed and vibrate in fixed positions. They have the least freedom of movement.
Liquid Phase (Water): As the temperature of ice increases, the energy supplied to the water molecules allows them to break free from their ordered arrangement and move more freely. In the liquid phase, water molecules have more freedom of movement compared to the solid phase. They can slide past one another and flow, taking the shape of the container they occupy.
Gas Phase (Water Vapor): At even higher temperatures, the energy imparted to the water molecules becomes sufficient to overcome the intermolecular forces holding them together. As a result, the water molecules separate and move independently in the form of a gas. In the gas phase (water vapor), the water molecules have the highest level of freedom of movement. They move rapidly and occupy the entire space available to them.
To know more about substance,
https://brainly.com/question/21782246
#SPJ11
There are 2 blue crayons and 6 red crayons. How
many fewer blue crayons are there?
Answers
Answer:
4
Step-by-step explanation:
6 - 2 = 4
Answer: 4
4
Explanation
6-2=4
help find the answer for a and b worth 10 points
Answers
Answer:
a) 2a + 6(a - 2) = 44
b) short side = 7 cm long side = 15 cm
Step-by-step explanation:
a) (2 x a) + [2 x 3(a - 2)] = 44
2a + 6(a - 2) = 44
b) 2a + 6(a - 2) = 44
2a + 6a - 12 = 44
8a -12 = 44
8a = 56
a = 7
Therefore, one side has length of 7 cm, and the other has
length of 3(7 - 2) = 15 cm.
Hope this helps!
en cuál de los siguientes procedimientos se resuelven sin error las siguiente operaciones?
Answers
D. \(\frac{(8)\cdot (-7)\cdot (6)}{(-4)\cdot (-2)\cdot (-3)} = \frac{-336}{-24} = 14\)
Por álgebra, recordamos los siguientes dos teoremas llamados teorema del signo para productos:
\((-x)\cdot y = x\cdot (-y) = -x\cdot y\), \(\forall \,x,y\in \mathbb{R}\) (1)
\((-x)\cdot (-y) = x\cdot y\), \(\forall \,x,y\in \mathbb{R}\) (2)
Basados en la información dada en la imagen y los teoremas descritos arriba, el producto del numerador debe ser negativo y el producto del numerador debe ser también negativo, más el resultado de la división debe ser positiva.
Por tanto, la opción D representa a la operación libre de errores.
Debemos recordar la regla de los signos:
(+)×(+) = (+)
(+)×(-) = (-)
(-)×(+) = (-)
(-)×(-) = (-)
Veremos que la opción correcta es la D.
Queremos resolver la expresión:
\(\frac{8*(-7)*(6)}{(-4)*(-2)*(-3)}\)
Primero debemos ver que hay 3 números en el numerador, de los cuales uno es negativo, por lo que el producto del numerador sera negativo:
Similar pasa para el denominador. El producto de los dos primeros dara un número positivo, y el producto de ese número positivo y el otro negativo dara un número negativo:
\(\frac{8*(-7)*(6)}{(-4)*(-2)*(-3)} = \frac{-336}{-24}\)
Ahora tenemos el cociente de dos números negativos, por lo que el cociente dara un número positivo:
\(\frac{8*(-7)*(6)}{(-4)*(-2)*(-3)} = \frac{-336}{-24} = 14\)
Así vemos que la opción correcta es D.
Si quieres aprender más, puedes leer:
https://brainly.com/question/24585838
a. for any equation containing the variables x and y, the derivative dy/dx can be found by first using algebra to rewrite the equation in the form yf(x). true or false
Answers
The statement "For any equation containing the variables x and y, the derivative dy/dx can be found by first using algebra to rewrite the equation in the form y = f(x)." is true
For any equation containing the variables x and y, the derivative dy/dx can be found by first using algebra to rewrite the equation in the form y=f(x).
To find the derivative dy/dx, we need to have the equation in the form y = f(x).
By rewriting the equation in this form using algebra,
we can then differentiate the function f(x) with respect to x to find the derivative dy/dx.
Learn more about dy/dx by implicit differentiation: https://brainly.com/question/31401678
#SPJ11
12. A five-question multiple-choice quiz has five choices for each answer. What is the
probability of correctly guessing at random exactly one correct answer? Exactly two
correct answers? Exactly three correct answers? (Hint: You could let any two digits
represent correct answers, and the other digits represent wrong answers.)
Answers
Answer:
20%
Step-by-step explanation:
what is the solution of 10(w +3) = 70
Answers
Answer:
w = 4
Step-by-step explanation:
distribute 10(w+3)=70
10w + 30 = 70
subtract 30
10w = 40
divide by 10
w = 4
Answer: w=7
Step-by-step explanation: 10 times what equals 70? 10 times 7 equals 10. So 3 plus 4 equals 7.
10(7+3)=70