Use Lagrange multipliers to find the indicated extrema, assuming that x, and y are positive.Minimize f(x,y)=x2−10x+y2−14y+28; Constraint: x+y=14
Answers
Answer:
- The values of x and y that minimize the function, subject to the given constraint are 6 and 8 respectively.
- The minimum value of the function = -44
Step-by-step explanation:
The Lagrange multiploer method finds the optimum value of a multivariable function subjected to a given constraint
It replaces the function with a Lagrange equivalent which is
L(x, y) = F(x, y) - λ C(x, y)
where λ Is the lagrange multiplier which can be a function of x and y
F(x, y) = x² - 10x + y² - 14y + 28
C(x, y) = x + y - 14
L(x, y) = x² - 10x + y² - 14y + 28 - λ (x + y - 14)
We now take the partial derivatives of the Lagrange function with respect to x, y and λ respecrively. Then solving to obtain values of x, y and λ that correspond to the minimum of the function. Since the first partial derivatives are all equal to 0 at minimum point.
(∂L/∂x) = 2x - 10 - λ = 0 (eqn 1)
(∂L/∂y) = 2y - 14 - λ = 0 (eqn 2)
(∂L/∂λ) = x + y - 14 = 0 (eqn 3)
Equating eqn 1 and 2
2x - 10 - λ = 2y - 14 - λ
2x - 10 = 2y - 14
2y = 2x - 10 + 14
2y = 2x + 4
y = x + 2 (eqn *)
Substitute eqn ^ into eqn 3
x + y - 14 = 0
x + x + 2 - 14 = 0
2x - 12 = 0
2x = 12
x = 6
y = x + 2 = 6 + 2 = 8
2x - 10 - λ = 0
12 - 10 - λ = 0
λ = 2
The values of x and y that minimize the function are 6 and 8 respectively.
F(x, y) = x² - 10x + y² - 14y + 28
At minimum point, x = 6, y = 8
F(x, y) = 6² - 10(6) + 8² - 14(8) + 28 = 36 - 60 + 64 - 112 + 28 = -44
Hope this Helps!!!
Related Questions
In some division problems, a number or pattern of number that continues indefinitely is a?
S = the team's scoreg = the number of goals a team has scoredWhich of the variables is independent and which is dependent?
What is the value of (-3.2)(- 1.5) ?
I can’t find the answer tried twice probably easy for others
Answers
3.2 + 36 - 2 * 7.2
3.2 + 36 - (-14.4)
3.2 + 36 + 14.4
24.8
24.8 is your answer.
B: 751.29
C: 1954.13
D: 1536.19
Solution:
use formula P[((1+(r/n)^(nt))-1)/(r/n)]
Solution 50[((1+(0.48/12)^(2 x 12))-1)/(0.48/12)]
= C $1954.13
Answers
It is given that you have invested $50 a month in an annuity that earns 48% APR compounded monthly. We can conclude that after 2 years you will have $1954.13 in your account.
How to solve future value?
To solve this we are going to use the formula for the future value of an ordinary annuity:
where
FV is the future value
P is the periodic payment
r is the interest rate in decimal form
n is the number of times the interest is compounded per year
t is the number of years
It is given that you have invested $50 a month in an annuity that earns 48% APR compounded monthly. we need to find how much money you have in this account after 2 years.
Since the interest is compounded monthly, it is compounded 12 times per year; therefore,
r = 48% = 0.48
n = 12
Let's put the values in our formula:
Thus, We can conclude that after 2 years you will have $1954.13 in your account.
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Answers
Answer:
14
Step-by-step explanation:
2^3=8
6^1=6
8+6=14
Answer:
2^3+6^1
8+6
14
Step-by-step explanation:
2^3 can be expanded to 2x2x2 which equals 8
6^1 can be expanded to 6x1 which is 6
2^3+6^1 can be simplified to to 8+6
8+6 is 14
B. The outcome could easily occur by chance.
C. The likelihood of getting these results by chance is very small.
D. The results do not make enough difference to be of use.
Answers
Answer:
C. The likelihood of getting these results by chance is very small.
Step-by-step explanation:
When the finding of a statistical analysis of data is said to be statistically significant, it simply means that the likelihood of getting these results by chance is very small.
This ultimately implies that, an experiment or research is statistically significant when its results or outcomes isn't likely to occur by chance or randomly rather it is associated with a specific cause.
Generally, when the p-value is very small (less than or equal to 0.05) then the result of a research or experiment is considered to be statistically significant and reliable.
Final answer:
Statistical significance in statistical analysis refers to the low likelihood of the identified result or relationship occurring by chance. It implies a high degree of certainty in the findings.
Explanation:
For the finding of a statistical analysis of data to be statistically significant, it essentially means that the likelihood of obtaining these results purely by chance is very low, making option C the correct one. In other words, when we say a result is statistically significant, we're confident that it’s not caused just by random chance but that there’s an underlying relationship at play. For instance, if a drug trial shows a significant difference in outcomes between a control group and a treatment group, it means there's a high likelihood that the drug is effective, not that the results occurred just by chance.
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Answers
The diagonals bisect each other (bisect means cut in half)
The diagonals are perpendicular
x1 = 172 x2 = 654
Answers
Answer:
The calculated value Z = 3.775 > 1.96 at 0.05 level of significance
Null hypothesis is rejected
The Two Population proportion are not equal
Step-by-step explanation:
Given first sample size n₁ = 677
First sample proportion
Given second sample size n₂ = 3377
second sample proportion
Null Hypothesis : H₀ : p₁ = p₂.
Alternative Hypothesis : H₁ : p₁ ≠ p₂.
Test statistic
where
P = 0.2036
Q = 1 - P = 1 - 0.2036 = 0.7964
Z = 3.775
Critical value ∝=0.05
Z- value = 1.96
The calculated value Z = 3.775 > 1.96 at 0.05 level of significance
Null hypothesis is rejected
The Two Population proportion are not equal