Compute the gradient of [tex]f[/tex].
[tex]\nabla f(x,y) = \left\langle 3x^2 - 6y, -6x + 3y^2\right\rangle[/tex]
Set this equal to the zero vector and solve for the critical points.
[tex]3x^2-6y = 0 \implies x^2 = 2y[/tex]
[tex]-6x+3y^2=0 \implies y^2 = 2x \implies y = \pm\sqrt{2x}[/tex]
[tex]\implies x^2 = \pm2\sqrt{2x}[/tex]
[tex]\implies x^4 = 8x[/tex]
[tex]\implies x^4 - 8x = 0[/tex]
[tex]\implies x (x-2) (x^2 + 2x + 4) = 0[/tex]
[tex]\implies x = 0 \text{ or } x-2 = 0 \text{ or } x^2 + 2x + 4 = 0[/tex]
[tex]\implies x = 0 \text{ or } x = 2 \text{ or } (x+1)^2 + 3 = 0[/tex]
The last case has no real solution, so we can ignore it.
Now,
[tex]x=0 \implies 0^2 = 2y \implies y=0[/tex]
[tex]x=2 \implies 2^2 = 2y \implies y=2[/tex]
so we have two critical points (0, 0) and (2, 2).
Compute the Hessian matrix (i.e. Jacobian of the gradient).
[tex]H(x,y) = \begin{bmatrix} 6x & -6 \\ -6 & 6y \end{bmatrix}[/tex]
Check the sign of the determinant of the Hessian at each of the critical points.
[tex]\det H(0,0) = \begin{vmatrix} 0 & -6 \\ -6 & 0 \end{vmatrix} = -36 < 0[/tex]
which indicates a saddle point at (0, 0);
[tex]\det H(2,2) = \begin{vmatrix} 12 & -6 \\ -6 & 12 \end{vmatrix} = 108 > 0[/tex]
We also have [tex]f_{xx}(2,2) = 12 > 0[/tex], which together indicate a local minimum at (2, 2).
What is the measure of arc ECF in circle G? pls show work, I don't get edge teachers
The measure of arc ECF in circle G is 98°. The correct option is the second option 98°
Circle GeometryFrom the question, we are to determine the measure of arc ECF
By using the one of the circle theorems
The angle subtended by an arc of a circle is twice the angle it subtends anywhere on the circle's circumference
In the given diagram,
∠F = 1/2 ×104°
∠F = 52°
∠D + ∠E + ∠F = 180° (Sum of angles in a triangle)
∠D + 79° + 52° = 180°
∠D = 180° - 79° - 52°
∠D = 49°
m arc ECF = 2 × m ∠D
m arc ECF = 2 × 49°
m arc ECF = 98°
Hence, the measure of arc ECF in circle G is 98°. The correct option is the second option 98°
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Vernon Taber deposits $600 into an ordinary annuity after each quarter for 4 years. The account pays 4% interest compounded quarterly. Find the a) future value, and b) total interest earned.
The future value is $10,354.72.
The total interest earned is $754.72.
What is the future value and the interest earned?When an amount is compounded quarterly, it means that both the amount deposited and interest already earned increases in value every quarter.
The future value = quarterly deposit x annuity factor
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest rate = 4% / 4 = 1%n = number of years = 4 x 4 = 16Annuity factor = [1.01^16) - 1 ] / 0.01 = 17.26
17.26 x 600 = $10,354.72
Total interest earned = future value - total amount deposited
Total amount deposited = $600 X 4 X 4 = 9600
$10,354.72 - 9600 = $754.72
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What is the range of the averages in this class? histogram group of answer choices 65 points 80 points 30 points none of the above
The range of the averages in a class with a maximum point of 100 point and a minimum point of 20 point is 80 points
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Range is the difference between the maximum value and minimum value of a data set. Hence:
Range = Maximum - Minimum
Let us assume that the maximum point is 100 point and the minimum point is 20 point, hence:
Range = 100 - 20 = 80 points
The range of the averages in a class with a maximum point of 100 point and a minimum point of 20 point is 80 points
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A bus travels from Boston to New York 220 miles. The bus departs at 2pm and arrives at new York at 7pm. what's the average speed?
Write an equation of the line passing through the point (6,4) that is perpendicular to the line y=- 1/2x +4
Hello,
"perpendicular to the line y= - 1/2x +4 "
so y = -1/2x + p
-1/2 × 6 + p = 4 ⇔ -3 + p = 4 ⇔ p = 4 + 3 = 7
y = -1/2x + 7
In stats would a test looking at gender & holiday preference would I do a Chi-squared test? Which is the explanatory & which is the response variable?
Any help provided is appreciated. Thanks!
In statistics, a Chi-squared test may be used to determine holiday choice and gender and α (alpha) is the response variable.
What is the Chi-squared test?A statistical technique called the chi-square test is used to compare actual outcomes with predictions.
The goal of this test is to establish if a discrepancy between actual and predicted data is the result of chance or a correlation between the variables you are researching.
Whether there is a statistically significant association between categorical variables is determined by the Chi-square test of independence.
This issue is addressed by a hypothesis test. The chi-square test of association is another name for this assessment.
Hence,in stats would a test looking at gender & holiday preference yes you can do a Chi-squared test and α(alpha) is the response variable.
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Find the exponential function that contains the points (3,24) and (4,48).
The required exponential function that contains the points (3,24) and (4,48) is y = 3(2)^x
Exponential functionsThe standard exponential function is expresses as y = ab^x
Given the coordinate points (3,24) and (4,48), the equivalent equations are:
24 = ab^3 ......... 1
48 = ab^4
Take the ratios
48/24 = b^4/b^3
2 = b
Substitute b = 2 into equation 1
24 = a(2)^3
8a = 24
a = 3
Hence the required exponential function that contains the points (3,24) and (4,48) is y = 3(2)^x
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Write an expression that is equal to 20 usong only one 2 and two 4s and any number of math symbols
Answer:
4^2+4
Step-by-step explanation:
one 2 and two 4s
so 4^2+4=16+4=20
so 4^2+4
(calculation)
wo competing companies are having sales on projectors. Projector #MD375 has a retail price of $1,200. The Office Spot is advertising #MD375 for 20% off, while Office & Beyond has stated they will match any competitor’s advertised special and take an additional 15% off of the competitor’s advertised price. How much money is saved by purchasing the projector from Office & Beyond instead of the Office Spot?
The amount of money that is saved by purchasing the projector at Office and beyond is $144.
How much money is saved?The first step is to determine the price of the projector at The Office spot.
Discounted price : (1 - 0.20) x $1200
0.80 x $1200 = $960
The second step is to determine the price of the projector at the Office and Beyond.
Discounted price : (1 - 0.15) x $960 = $816
Difference in price : 960 - 816 = $144
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In survey of 100 students in the institute of Language Studies at Addis Ababa University,
the numbers of students studying various languages were found to be English 28,
Amharic 30, Afan Oromo 42, English and Afan Oromo 10, English and Amharic 8,
Amharic and Afan Oromo 5, and all the three languages 3. How many students
a. were studying none of the languages?
b. had Afan Oromo as their only language?
c. studied Amharic if they studied Afan Oromo?
Answer:
a. 17
b. 30
c. 5
Step-by-step explanation:
English (E) = 28 - 15 (10+8-3) = 13
Amharic (A) = 30 - 10 (8+5-3) = 20
Afan Oromo (AO) = 42 - 12(10+5-3) = 30
E+AO = 10
E + A = 8
A + AO = 5
A + AO + E = 3
a. 13 people spoke just English+ 20 people who just spoke Amharic + 30 people who just spoke Afan Oromo+ The multilingual people (10+8+5-3=20) which is 83, so that means that 17 learned no language.
b. 42 People learned Afan Oromo, but only 30 of them learned that language separately. 10 People learned it along with English, and 5 learned it with Amharic. However, 3 of those people were the same, so 12 people of those 42 spoke multiple languages. Therefore, only 30 people spoke just Afan Oromo.
c. This one is kinda just a trick question, they want you to overthink it because of the 3 who learned all, but it's just 5 like it says in the question.
There's 18 deer and 52 lions if 10 lions eat 17 deer how many deer are left
Answer:
1 deer
Step-by-step explanation:
18 deer minus 17 eaten leaves 1 deer
Which of the following examples illustrates ordinal numbers?
A. Marie counted the number of chips on her plate and said, " I have eight".
B. Gracie told Bradly she would be the first one to go down the slide.
C. Bennett said he had more blocks than anyone.
Answer:
30-5×2 of 3+(19-3) ÷8
Step-by-step explanation:
30-5×6+(19-3)÷8
30-5×6+16÷8
30-30+8
38-30
8
Answer:
Step-by-step explanation:
b
A train travels 65 miles and then stops in a city. Then it travels at a speed of 95 miles per hour before reaching its last stop.
Enter a linear model which represents the total distance traveled, d, by the train, as a function of t, the number of hours
after leaving the city
Step-by-step explanation:
as speed is always
distance/time
we need to multiply speed by the time traveled to get the distance traveled.
so,
d = 95t + 65
after all, until it reached the city it traveled already 65 miles (we don't know the speed, but it does not matter for the question about the total distance traveled), which we need to add to get the total distance.
how do i factor f(x)= x^4 - 4x^3 +3x^2 + 4x -4 into linear terms?
The factor of the polynomial f(x) = x⁴ - 4x³ + 3x² + 4x -4 are x = -1, x = 1 and x = 2.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the function f(x) = x⁴ - 4x³ + 3x² + 4x -4. The factor of the polynomial f(x) = x⁴ - 4x³ + 3x² + 4x -4 are x = -1, x = 1 and x = 2.
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What is the solution set for this inequality? 5x − 4 < 11 + 8x
Answer: x>-5
Step-by-step explanation:
add 4 to both sides
5x-4+4<11+8x+4
simplify
5x<8x+15
subtract 8x from both sides
5x-8x<8x+15
simplify
-3x<15
divide both sides by -3 and switch inequality sign
-3x/-3>-15/-3
x>-5
[tex]\large\displaystyle\text{$\begin{gathered}\sf 5x-4 < 11+8x \end{gathered}$}[/tex]
Simplify both sides of the inequality.
[tex]\large\displaystyle\text{$\begin{gathered}\sf 5x-4 < 8x+11 \end{gathered}$}[/tex]
Subtract 8x from both sides.
[tex]\large\displaystyle\text{$\begin{gathered}\sf 5x-4-8x < 8x+11-8x \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf -3x-4 < 11 \end{gathered}$}[/tex]
Add 4 to both sides.
[tex]\large\displaystyle\text{$\begin{gathered}\sf -3x-4+4 < 11+4 \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf -3x < 15 \end{gathered}$}[/tex]
Divide both sides by -3.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{-3x}{-3} < \frac{15}{-3} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf x > -5 \ \ \iff \ \ \ Answer \end{gathered}$}[/tex]
{ Pisces04 }If f(x) = x2 + x and g(x) = x − 3, find f(g(7)).
The value of f(g(7)) is 20
How to determine the composite functions?The functions are given as:
f(x) = x^2 + x
g(x) = x - 3
Start by calculating g(7)
g(7) = 7 - 3
Evaluate
g(7) = 4
So, we have:
f(g(7)) = f(4)
Substitute 4 for x in f(x) = x^2 + x
f(4) = 4^2 + 4
Evaluate
f(4) = 20
Hence, the value of f(g(7)) is 20
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Solve the mathematical puzzle.
Determine the digits of Y
from these clues.
The digits of Y
add to 16.
The first digit is 4 times the third digit.
The second digit is 3 times the third digit.
Y is a three digit number.
Answer:
862
Step-by-step explanation:
4 x 2 = 8
3 x 2 = 6
8 + 6 + 2 = 16
A convenience store recently started to carry a new brand of soft drink. Management is interested in estimating future sales volume to determine whether it should
continue to carry the new brand or replace it with another brand. The following table provides the number of cans sold per week. Use both the trend projection with
regression and the exponential smoothing (let a=0.4 with an initial forecast for week 1 of 574) methods to forecast demand for week 13. Compare these methods by
using the mean absolute deviation and mean absolute percent error performance criteria. Does your analysis suggest that sales are trending and if so, by how much?
2
5
Period 1
3 4
574 635 652 739 679
Sales
with regression forecast.
(Enter your response rounded to the nearest whole number.)
(1) Obtain the trend projection
The forecast for week 13 is
6
637
7
723
8
691
9
773
10
718
11
656
12
729
The forecast for week 13 is 9.7%
a) MAD without the forecast = 43.71
MAD with the forecast = 45.59
b) MAPE = 0.097
MAPE= 9.7%
642 602 656 747 663 618 731 726 679 737 664 740
Mean Absolute deviation is given as
MAD = [Σ|x - μ|]/N
We first calculate the mean
Mean = Σx/N
Mean = (642+602+656+747+663+618+731+726+679+737+664+740)/12
Mean = (8205/12)
Mean= 683.75
absolute deviations from the mean
|642-683.75| + |602-683.75| + |656-683.75| + |747-683.75| + |663-683.75| + |618-683.75| + |731-683.75| + |726-683.75| + |679-683.75| + |737-683.75| + |664-683.75| + |740-683.75|
Absolute deviations from the mean
41.75+81.75+27.75+63.25+20.75+65.75+47.25+42.25+4.75+53.25+19.75+56.25= 524.5.
What is the formula for absolute mean deviation?MAD = [Σ|x - μ|]/N
MAD = (524.5/12)
MAD = 43.71
If we include the forecast for the 13th week
642 602 656 747 663 618 731 726 679 737 664 740 747
Mean = (8205+747)/13
Mean = 688.62
absolute deviations from the mean
|642-688.62| + |602-688.62| + |656-688.62| + |747-688.62| + |663-688.62| + |618-688.62| + |731-688.62| + |726-688.62| + |679-688.62| + |737-688.62| + |664-688.62| + |740-688.62| + |747-688.62|
Absolute deviations from the mean
=46.62+86.62+32.62+58.38+25.62+70.62+42.38+37.38+9.62+48.38+24.62+51.38+58.38
= 592.62
MAD = (592.62/13)
MAD = 45.59.
b) MAPE is used to check forecast errors
MAPE = (1/N) Σ [|x-f|/x]
where N = Sample size = 12
x = each variable
f = the forecasted value = 747
x = 642, 602, 656, 747, 663, 618, 731, 726, 679, 737, 664, 740
Σ[|xf|/x]=0.164+0.241+0.139+0+0.112+0.209+0.022+0.029+0.100+0.014+0.125+0.009
Σ [|x-f|/x] = 1.164
MAPE = (1.164/12)
MAPE = 0.097
MAPE=9.7%
The forecast for week 13 is 9.7%.
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A particular sawdust pile has a base diameter of 25 feet when the height is 20 feet. Find the volume of this sawdust pile when it is 30 feet high.
The volume of the sawdust pile is 11,039.06 ft³.
What is the volume of the pile?The shape of the sawdust pile would be a cone. The diameter is always proportional to its height.
Diameter = k x height
where k is equivalent to the rate of increase of the diameter with respect to the increase in height.
25= k20
k = 25 / 20
k = 1.25
Diameter when height is 30 ft = 1.25x 30 = 35.50 feet
Volume of the sawdust = [tex]\frac{1}{3}[/tex]πr²h
Where:
π = 3.14 r = radius = diameter / 2 = 35.50 / 2 = 18.75h = 30[tex]\frac{1}{3}[/tex] x 3.14 x 18.75² x 30 = 11,039.06 ft³
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Problem 10-09 (Algorithmic) Cress Electronic Products manufactures components used in the automotive industry. Cress purchases parts for use in its manufacturing operation from a variety of different suppliers. One particular supplier provides a part where the assumptions of the EOQ model are realistic. The annual demand is 5,300 units, the ordering cost is $70 per order, and the annual holding cost rate is 25%.
The economic order quantity is 385 and reorder point is 254.
Economic order quantity (EOQ)a. Economic order quantity (EOQ)
EOQ=√2×D×C÷H
Where:
D =Annual Demand
S = Cost of Ordering
H=Holding cost
Hence:
EOQ=√2×5300×70÷.25×20
EOQ=√742,000÷5
EOQ=√148,400
EOQ=385
b. Reorder point
Reorder point=5300/250×12 days
Reorder point=254
c. Reorder point
Reorder point=5300/250×35 days
Reorder point=742
d. Reorder point
Reorder point=742-385
Reorder point=357
Therefore the economic order quantity is 385 and Reorder point is 254.
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The complete question is:
a. If the cost of the part is $20 per unit, what is the economic order quantity?
b. Assume 250 days of operation per year. If the lead time for an order is 12 days, what
is the reorder point?
c. If the lead time for the part is seven weeks (35 days), what is the reorder point?
d. What is the reorder point for part (c) if the reorder point is expressed in terms of the inventory on hand rather than the inventory position?
2
Drag each number or expression to the correct location on the table. Each number or expression can be used more than once, but not all
numbers or expressions will be used.
Complete the empty cells in the table by determining all terms, constants, and coefficients for both algebraic expressions.
-5 7 5
Expressions
Terms
Constants
Coefficients
-5a
11
Reset
4a2
4a2. -5a-3
Next
7(-3x + 5) + 1ly
7(-3x + 5)
11y
5
The empty cells in the table by determining all terms, constants, and coefficients for both algebraic expressions will be:
Constant = -3Coefficient = -4, -5How to illustrate the information?In the first expression, the equation is given as 4a² - 5a - 3 while in the second expression, the equation is given as 7(-3x + 5) + 11y.
In this second expression, the term is 7(3x + 5), 11y. The constants are -7 and 5. Lastly, the coefficients are -3, and 11.
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Each side of a hexagon is 10 inches longer than the previous side. What is the length of the shortest side of this hexagon if its perimeter is 401 inches?
Answer:
41.83 inches
Step-by-Step Explanation:
You have to trust me. I helped my brother with this.
The length of the shortest side of the hexagon is approximately 41.83 inches
Let's denote the length of the shortest side of the hexagon as "x" inches. According to the given information, each side of the hexagon is 10 inches longer than the previous side.
So, the lengths of the sides of the hexagon can be expressed as follows:
First side: x inches
Second side: (x + 10) inches
Third side: (x + 20) inches
Fourth side: (x + 30) inches
Fifth side: (x + 40) inches
Sixth side: (x + 50) inches
To find the perimeter of the hexagon, we sum up the lengths of all the sides:
Perimeter = x + (x + 10) + (x + 20) + (x + 30) + (x + 40) + (x + 50)
We know that the perimeter is given as 401 inches, so we can set up the equation:
401 = x + (x + 10) + (x + 20) + (x + 30) + (x + 40) + (x + 50)
Simplifying the equation:
401 = 6x + 150
Subtracting 150 from both sides:
251 = 6x
Dividing both sides by 6:
x = 251/6
Therefore, the length of the shortest side of the hexagon is approximately 41.83 inches (rounded to two decimal places).
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How many 4-character passwords can you generate if the first character can be any letter of the English alphabet, the next two characters can be any two digits from 0 to 9, and the last one can be any of the characters %,*, or #? (Repetition is allowed!)
There are 7,800 different passwords.
How many different passwords can be generated?
Here we need to count the number of options for each of the digits of the password.
The first character has 26 options (26 letters in the alphabet).
The second character has 10 options (ditis from 0 to 9), and also does the third character
The fourth character has 3 options (%,*, or #)
The total number of different passwords that can be generated is given by the product between the numbers of options, so we have:
C = 26*10*10*3 = 7,800
There are 7,800 different passwords.
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Which answer is correct?
The reason that would support that is ∠B ≅ ∠B , the reflective property of equality.
What are similar triangles?Similar triangles are triangles that have their corresponding angles equal and the ratio of their corresponding lengths also equal.
Analysis:
triangle BDE and BAC share the same angle B
which means ∠DBE = ∠ABC.
The reflective property of equality says a number is always equal to itself. Therefore, ∠B ≅ ∠B
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A sample of size 400 has mean =6.0. Can it be regarded as a sample from a large population with mean 6.25 inches and standard deviation 2.25 inches ? using α=0.05
The value of Z at the 5% level is not significant if the sample of size 400 has mean is 6.0.
What is the standard deviation?It is defined as the measure of data disbursement, It gives an idea about how much is the data spread out.
[tex]\rm \sigma = \sqrt{\dfrac{ \sum (x_i-X)}{n}[/tex]
σ is the standard deviation
xi is each value from the data set
X is the mean of the data set
n is the number of observations in the data set.
The standard error of the estimated mean:
[tex]\rm = \dfrac{2.5}{\sqrt{400}}[/tex]
= 0.1125
x = 6.0–6.2 = 0.2
[tex]\rm Z = \dfrac{0.2}{0.1125}[/tex]
Z = 1.778
This is not significant at the 5% level.
This would have been significant at the 5% level if the same sample means had been found using a sample size of merely 500.
Thus, the value of Z at the 5% level is not significant if the sample of size 400 has mean is 6.0.
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which expressions are equivalent to 3x
x is powering both numbers so it can be outside the parenthesis.
We have given that 3^x.
What is the expression?An expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
The first one isn't an answer because 3^x is exponential while x^3 is a cubic function.
If you draw them you will see that they are very different.
B is correct because we can divide both numerator and denominator by 6 and we get 3^x.
C is not correct because x is not powering 3 so we cannot divide both by 6D is correct because 3^(x-1) is the same as
and when multiplied by 3 we get 3^x
3^x*3^(-1) = 3^x/3
E is not correct.
will understand after the explanation in DF is correct.
x is powering both numbers so it can be outside the parenthesis.
The question is incomplete the complete question is,
Which expressions are equivalent to the one below? Check all that apply.
3^x
A. x^3
B.(18/6)^x
C.18^x/3
D.3(3^(x-1))
E.3(3^(x+1))
F.18^x/6^x
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Volume (to the nearest tenth)
Answer:
913
Step-by-step explanation:
cylinder:
radius=5.6 height=7
first lets find the volume of the cylinder the formula for the volume of a cylinder is pi times radius squared times height. so after plugging in the height and radius you get 689.64242
cone:
radius=5.6 height=6.8
now the volume of the cone, the formula for the volume of a cone is pi times radius squared times height/3. so after plugging in the height and radius of the cone you get 223.31278
so 689.64242 + 223.31278= 912.9552 and that rounded to the nearest tenth is 913
Study: Factoring Polynomials Completely
What are the possible roots of the polynomial function below? (Check all
that apply.)
F(x) = 2x² + 5x -5
Answer:
x = 0.76556443 x = -3.26556443
Step-by-step explanation:
You can use the quadratic formula to find the roots.
Simplify:
(12+14) x 8+ (8² +4²) =
Answer:
288
Step-by-step explanation:
[tex]26 * 8+(2^{6} +2^{4} )\\\\208+(64+16)\\208+80[/tex]
Answer:
288
Step-by-step explanation:
PEMDAS
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Parentheses and Exponents:
(26) x 8+(64+16) (28) x 8+(80)
Multiplication:
26 x 8 + 80 208+80
Addition:
208+80= 288
(PLEASE HELP FAST!!!) Which expression gives the area of triangle QRS?
The expression which gives the area of triangle QRS is [tex]\frac{1}{2} (9)(15)sin(20^{o} )[/tex].
We need to find the area of triangle QRS.
How do I find the area of a triangle using trigonometry?The area of a triangle can be expressed using the lengths of two sides and the sine of the included angle. Area of Δ = ½ ab sin C.
Now, the area of the triangle QRS[tex]=\frac{1}{2} (9)(15)sin(20^{o} )[/tex]
Therefore, the expression which gives the area of triangle QRS is [tex]\frac{1}{2} (9)(15)sin(20^{o} )[/tex].
To learn more about the area of a triangle using trigonometric ratios visit:
https://brainly.com/question/18659560.
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