Find the local extreme values of the function f(x, y) = xy - x2 - y2 - 3x - 3y + 12

Answers

Answer 1

[tex]f(x,y)=xy-x^2-y^2-3x-3y+12[/tex]

First compute the first-order partial derivatives and find the critical points.

[tex]f_x=y-2x-3[/tex]
[tex]f_y=x-2y-3[/tex]

Both first order derivatives vanish at [tex](x,y)=(-3,-3)[/tex].

Computing the Hessian, we get

[tex]\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}-2&1\\1&-2\end{bmatrix}[/tex]

We have [tex]\det\mathbf H(x,y)=3>0[/tex], which means [tex](-3,-3)[/tex] is an extremum of [tex]f(x,y)[/tex]. Since [tex]f_{xx}(-3,-3)=-2<0[/tex], this extremum is a local maximum of [tex]f(x,y)[/tex] with a value of 21.

Related Questions

cylinder A has a volume of 360 cm3. Cylinder B has a base and height identical to that of cylinder B, but leans to the right in such a way that its slant is greater than 4 cm. what is the volume of cylinder B

Answers

If the height is measured between the parallel ends of the cylinder, the volume is identical:
360 cm³.

_____
If the "height" is measured along the central axis of the cylinder, there is insufficient information to determine the volume.

The slant length of a prism (even a cylindrical one) is immaterial. What matters to the volume is the area of the base and the distance between the planes of the bases (height).

hey user

360

is the answer

To finance her community college education, Sarah takes out a loan for $3700. After a year Sarah decides to pay off the interest, which is 5% of $3700. How much will she pay?

Answers

So, to solve this, we use this equation:
3700 + 0.05(3700)
But, if we want to make it shorter:
1.05(3700)
Now you just multiply(you may want to use a calculator)
1.05(3700) = 3885
She will pay $3885

Use the distributive property to factor the expression below.

54x2y - 12x2y2

Answers

The given expression is:

[tex]54 x^{2} y-12 x^{2} y^{2} [/tex]

The term 6x²y can be factored out from both the terms, leaving behind the expression given below.

[tex]6 x^{2} y(9-2y)[/tex]
So,

[tex]54 x^{2} y-12 x^{2} y^{2} = 6 x^{2} y(9-2y)[/tex]

Q #11 Solve the equation

Answers

67% = 0.67
360*0.67 = 241.2
Your answer is 241°

For this case we can make the following rule of three:
100% --------> 360 degrees
67% ---------> x
Clearing the value of x we have:
x = (67/100) * (360)
x = 0.67 * 360
x = 241.2 degrees
Rounding off we have:
x = 241 degrees
Answer:
x = 241 degrees
option 4

Which type of study contains a treatment?

Sample.
Experiment.
Survey.
Observational study.

Answers

Typically, in an experiment there are two groups. One group is called the control group. The other group is the experimental group. This second group receives some sort of treatment. The control group does not. The idea is to see if the treatment has an effect on those who receive it by comparing them to a comparable group that does not receive the treatment.

So the answer is experiment.

The type of study that contains a treatment is called EXPERIMENTAL STUDY.
An experimental set up usually involved random assignment of variables into different group and manipulation of a particular variable while others are controlled. This type of study is usually done to determine the effect of one variable on the outcome of another variable. The experimental set up is usually made up of control and treatment groups.

David’s starting pay for his summer job is $10.00 per hour. Halfway through the summer he receives a 50 cent per hour raise. David is hired the next summer and receives a 10% raise. Halfway through the second summer he receives a 50 cent per hour raise. David is hired for a third summer and is given a 5% raise. How much money does David earn per hour during the third summer? Show or explain your work.

Answers

$10, then $10.50, 10% of that is 1.05 which would make it $11.55. Then $12.05, 5% of 12.05 is .60 which would make it $12.65.

The final answer is $12.65

Choose the single logarithmic expression that is equivalent to the one shown 5 log x + 3 log x^2

Answers

5 log x+3 log x^2
=log x^5+log x^(2×3)
=log x^5+ log x^6
=log (x^5×x^6)
=log x^(5+6)
=log x^11

Answer: log x^11

Nancy wove a pot holder with a area of 80 square inches. The lengths and widths of the sides are whole numbers. Which dimensions make the most sense for a potholder? Explain

Answers

The pairs of numbers that multiply up to 80 are (1, 80), (2, 40), (4, 20), (5,16), and (8,10).
Since pot holders are normally close to squares, 8 by 10 makes the most sense

Final answer:

The dimensions of 8 inches by 10 inches make the most sense for a pot holder with an area of 80 square inches, as they are practical in size. This is similar to the concept of a square whose area scales by the square of the side length when dimensions are increased by a scale factor.

Explanation:

The student asked which dimensions make the most sense for a potholder with an area of 80 square inches where the lengths and widths are whole numbers. It's crucial to consider practical sizes that would be convenient for handling hot items without being too large or too small.

Common dimensions for a pot holder could be 8 inches by 10 inches since 8 x 10 equals 80 square inches, and these measurements are suitable for a pot holder. These measurements make sense as they provide enough space to protect hands from heat but are not excessively large, making them impractical. They are comparable to the measurements of a placemat, which is typically used in dining settings and is also manageable in size.

Comparing this to the information about squares and areas, if Marta has a square with side length 4 inches, its area would be 16 square inches. A similar square with side lengths that are twice as long (8 inches) would have an area of 64 square inches because the area is proportional to the square of the side length. Consequently, the area of the larger square is four times that of the smaller square (since 2 squared is 4).

This example of scaling dimensions and areas can assist in understanding how dimensions that are whole numbers relate to the area and the practical use of an object. The mentioned scale factor is integral in determining how the area increases with increased side lengths.

3x^2-2x+7 divided by x+1 what is the quotient?

Answers

Dividend: D(x)= 3x^2-2x+7

Using Syntetic Division:

! 3 -2 7
Divisor: d(x)=x+1→-1 ! -3 5
----------------------
3 -5 12

Quotient: Q(x) = 3x -5

Answer: The quotient is 3x-5

3x^2-2x+7 divided by x+1 or

3x^2-2x+7 / x+1

By long division;

x+1 |3x^2-2x+7 = 3x-5

Hope this helps! Goodluck!

Choose any numbers for he first column that will correctly complete the table.

Answers

1. Can be any number between 1,765 - 1,769
2. Any number between 9,425 - 9,429

Number: 7,760

Number: 9,440

In order to accumulate enough money for a down payment on a house, a couple deposits $300/month into an account paying 6% compounded monthly. If payments are made at the end of each period, how much money will be in the account in 5 years? What formula is required to complete this problem?

Answers

[tex]\bf ~~~~~~~~~~~~\textit{Future Value of an ordinary annuity}\\ ~~~~~~~~~~~~(\textit{payments at the end of the period}) \\\\ A=pymnt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right][/tex]

[tex]\bf \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\\ pymnt=\textit{periodic payments}\to &300\\ r=rate\to 6\%\to \frac{6}{100}\to &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years\to &5 \end{cases}[/tex]

[tex]\bf A=300\left[ \cfrac{\left( 1+\frac{0.06}{12} \right)^{12\cdot 5}-1}{\frac{0.06}{12}} \right]\implies A=300\left[ \cfrac{(1.005)^{60}-1}{0.005} \right] \\\\\\ A=300(\stackrel{\approx}{69.770030509863214)}\implies A\approx 20931.009152958964[/tex]

A basket is filled with 8 white eggs and 10 brown eggs. Half of the white eggs are cracked. An egg is randomly selected from the basket. What is the probability that it is a cracked, white egg?Write the answer in a fraction in simplest form

Answers

Number of white eggs = 8
Number of brown eggs = 10
Total sample = 8+10 = 18

Number of broken eggs = 4 white

P(broken egg) = 4/18 = 2/9

PLEASE HELP!!!!!! WHICH OF THE FOLLOWING IS A CORRECT EQUATION FOR THE LINE PASSING THROUGH THE POINT (-2 1) AND HAVING SLOPE M = 1/2

Answers

The answer for this equation is A and B.

Answer choice A is in slope-intercept (y = mx+b) form. The slope of the line is the number in front of the x, so in this case it's [tex] \frac{1}{2} [/tex]. Now, just plug in the given point to see if the equation is satisfied:

[tex]1= \frac{1}{2} (-2)+2[/tex]
[tex]1= -1+2[/tex]
[tex]1= 1[/tex]

It works, so answer choice A works.

Answer choice B is in slope-intercept (y = mx+b) form. The slope of the line is the number in front of the x, so in this case it's -2. Answer choice B has the wrong slope, so eliminate B.

Answer choice C is in standard form. Rearrange it to slope-intercept form:

[tex]x-2y=-4[/tex]
[tex]-2y=-x-4[/tex]
[tex]y= \frac{1}{2} x+2[/tex]

Same as A, so answer choice C works.

Answer choice D is in point-slope form: y - y1 = m(x - x1). If you know this form well, you should be able to see that the given equation satisfies both the slope and point requirement. If you don't know the form well, try putting it into slope-intercept:

[tex]y-1= \frac{1}{2} (x+2) [/tex]
[tex]y-1= \frac{1}{2} x+1 [/tex]
[tex]y= \frac{1}{2} x+2 [/tex]

Same as A, so answer choice D works.

PLZ HELP ASAP CIRCLES

Answers

From the figure we are given that the measure of angle U is equal to 85°. With this, we can conclude that the arc intercepted by the angle which is the arc DKC will have a measure that is twice its value.

m(DKC) = 2(85°) = 170°

The total measure of the arc in a circle is equal to 360°. Hence,

m(DKC) + m(DUC) = 360°

Substituting the known value,
170° + m(DUC) = 360°

The value of m(DUC) is equal to 190°. The measure of the angle K is half this value.
m∠K = 190°/2 = 95°

Answer: 95°

Danny bought 5 candies. Lucy bought 2 fewer than danny. Jimmy bought 4 more than lucy. how many candies did jimmy buy

Answers

Jimmy bought 7 candies

Jimmy bought 7 candiss

a $1600 principal earns 7% annual interest, compounded semiannually. What is the balance after 33 years?

Answers

Use the future value formula.
FV = P(1 +r/n)^(nt)
FV = 1600*(1 +.07/2)^(2*33)
FV = 1600*1.035^66 ≈ 15,494.70

The balance after 33 years will be $15,494.70

Because your friends stayed in bed instead of helping you with the breakfast, you agree to split the £248 hostel bill to the ratio of: 3:3:2 with you paying the least. How much will you have to pay?

Answers

Final answer:

The question posed is about the concept of ratio, you are required to divide £248 according to the ratio 3:3:2. Each part of the ratio is worth £31, thus you will have to pay for two parts, hence, your total responsibility is £62.

Explanation:

The subject of this question is ratio. In the given problem, you need to distribute £248 hostel bill into the ratio of 3:3:2. Firstly, you add the three parts of the ratio which equals to 8 (3+3+2). Now, to find out how much each part is worth you need to divide the total sum, which is £248, by the total number of parts, which is 8. This gives you the value for one part, which is £31 (248 ÷ 8).

Given that you have to pay the least, which corresponds to 2 parts, you multiply £31 by 2, thus you have to pay £62 (31 * 2).

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A sales associate in a department store earns a commission on each suit and each pair of shoes sold. One week, she earned $47 in commission for selling 3 suits and a pair of shoes. The next week, she earned $107 in commission for selling 7 suits and 2 pairs of shoes. How much commission does she earn for selling each suit and each pair of shoes? Solve by substitution.

Answers

let h shoes and s suit:

3s+h=47

7s+2h=107

substitution property:

3s=47-h

s=(47-h)/3

plug it into the other equation and solve for h:

7((47-h)/3)+2h=107

distribute 7 and multiply 3 on both sides:

329-7h+6h = 321

329-h=321

-h=-8

h=8 pair of shoes

s=(47-8)/3=13 suits

hey can you please help me posted picture of question

Answers

G(x) is obtained from F(x) after both horizontal and vertical translation.

Following translations occur in F(x) to obtain G(x)
a) 2 units to Right
b) 4 units down

Translation towards right can be expressed as:

F(x - 2)

Vertically down translation can be expressed as:

F(x-2) - 4

So,

G(x) = (x-2)² - 4

Thus, option A is the correct answer.

using a table of value determine the solution to the equation below to the nearest fourth of a unit

2^(-x) + 1 = 5^x + 2

x= -1.50
x= 2.50
x= -0.50
x= 0.75

Answers

For finding the solution to something like this, it makes sense to write a function of x that is zero at the solution value. That function might be obtained by subtracting the right side from the equation.
f(x) = (2^-x +1) - (5^x +2)
Now, we want to find x so that f(x) = 0.

For the offered choices, we get
f(-1.50) ≈ 1.74
f(-0.50) ≈ -0.03
f(0.75) ≈ -3.75
f(2.50) ≈ -56.72

The obvious choice is
x = -0.50

One state leads the country in tart cherry production, producing 80 out of every 100 tart cherries each year. a. What percent of tart cherries are produced in this state?

Answers

So, to turn this into a percentage, we have to turn this into a decimal by dividing:
80/100 = 0.80
80% of tart cherries are produced in this state.

Given the equation f(x) = (x + 3)2 – 8, fill in the blanks:

Axis of symmetry: x =

Vertex:( , )

Number of real solutions:

Answers

Quadratic f(x) = (x -h)² +k has vertex (h, k) and axis of symmetry x=h. When k is negative, the number of real solutions is 2, because both branches of the function cross the x-axis.

In your equation, h = -3 and k = -8.

Axis of symmetry: x = -3
Vertex: (-3, -8)
Number of real solutions: 2

Final answer:

This detailed answer explains the axis of symmetry, vertex, and number of real solutions for the given quadratic equation.

Explanation:

Given the equation f(x) = (x + 3)² – 8

Axis of symmetry: x = -3

Vertex: (-3, -8)

Number of real solutions: 1

Indicate the general rule for this sequence. Find a 1 and the common difference. a 4 = 18 a 7 = 9

Answers

Final answer:

The general rule for the sequence is an = -9 + (n-1)(9). The first term (a1) is -9 and the common difference is 9.

Explanation:

Sequence Analysis:

To find the general rule for a sequence, we need to determine the pattern or relationship between the terms. In this case, we are given a4 = 18 and a7 = 9. To find the common difference, we subtract the two terms: 18 - 9 = 9. So, the common difference is 9. To find a1, we can use the formula: an = a1 + (n-1)d, where n is the term number and d is the common difference. Substituting n = 4, a4 = 18, and d = 9, we can solve for a1. 18 = a1 + (4-1)(9), which gives us a1 = 18 - 3(9) = 18 - 27 = -9. Therefore, the general rule for this sequence is an = -9 + (n-1)(9). The first term (a1) is -9 and the common difference is 9.

a^1 = 27

d = -3

Step-by-step explanation:

A cat and a dog have a race. The cat’s strides are 30% shorter than the dog’s but it makes 30% more strides than the dog. Which of them will win the race?

Answers

Asked and answered elsewhere.
https://brainly.com/question/9994969

A basketball is thrown with an initial upward velocity of 30 feet per second from a height of 6 feet above the ground. The equation h=-16t^(2)+30t+6 models the height in feet t seconds after the basketball is thrown. After the ball passes its maximum height, it comes down and then goes into the hoop at a height of 10 feet above the ground. About how long after it was thrown does it go into the hoop?

Answers

For this case we have the following equation:
h = -16t ^ (2) + 30t + 6
We substitute the value of h = 10 in the equation:
10 = -16t ^ (2) + 30t + 6
Rewriting we have:
0 = -16t ^ (2) + 30t + 6-10
0 = - 16t ^ (2) + 30t - 4
We look for the roots of the polynomial:
t1 = 0.144463904
t2 = 1.730536096
"the ball passes its maximum height, it comes down and then goes into the hoop". Therefore, the correct root is:
t = 1.730536096
Answer:
It goes into the hoop after:
t = 1.73 seconds

median mode mean and range on 84 45 10 28 75 63 16 55

Answers

PUT THIS SET IN ORDER FROM LEAST TO GREATEST:

10, 16, 28, 45, 55, 63, 75, 84
__________________
MEDIAN: Find the number in the middle of the set.
10, 16, 28, 45, 55, 63, 75, 84
There cannot be two medians, so you find the number between 45 & 55.
MEDIAN = 50
_________________
MEAN: Find the average by adding up all of the numbers and dividing them by how many numbers there are.
10 + 16 + 28 + 45 + 55 + 63 + 75 + 84 = 376
376/8 = 47
MEAN = 47
________________
MODE: The number that occurs most often. If no number repeats, there's no mode.
THERE IS NO MODE
________________
RANGE: The difference between the largest and smallest value in the set. Subtract the largest (84) and smallest (10) value.
84 - 10 = 74
RANGE = 74

Mean: 47. Mean is when you add up all the numbers, then divide by how many numbers you added (adding them together you get 376, which you divide by 8).

Median: 50. Median is when you sort every number numerically and find which number is in the middle. Since there are 8 numbers, there are 2 numbers you'd find the number equally in between each of those 2 numbers. That number being 50.

Mode: There is no mode. Mode is the number that is "most frequent" or is repeated the most in the number sequence.

Range: 74. Range is when you subtract the smallest number of the sequence from the largest number.

I hope this helps!

An equilateral triangle's perimeter and a rectangle's perimeter have the same numerical value. The length of the rectangle is the same as the side of the triangle. The width of the rectangle is 12. What is the length of the rectangle?

Answers

just set the perimeter equation equal to each other and put the values in that you know and solve

Good Treads and Speed Tracks manufacture tires. Good Treads claims that their tires last longer than Speed Tracks tires. A car company decided to test Good Tread's claim. They purchased 20 sets of Good Tread's tires and 20 sets of Speed Track's tires. They tested the lifetime of each tire by simulating 1,000 miles of driving conditions for each set of tires each day. At the end of each day, they measured the tread on each of the tires. Once the tire tread drops below minimum safe tread depth, the tire is considered to have passed its lifetime.

After observing the lifetime of each tire, the car company reported that the average lifetime of a Good Treads tire is 55,000 miles and the average lifetime of a Speed Tracks tire is 65,000 miles.

Given the car company's report, what conclusion can be made regarding the average lifetime of Good Treads and Speed Tracks tires?
There is not enough information to make any conclusions from the car company's report.

Good Treads' claim is false; therefore, it can be concluded that Speed Tracks tires last longer than Good Treads tires.

Good Treads' claim is false; therefore, it can be concluded that Good Treads tires last longer than Speed Tracks tires.

Good Treads' claim is true; therefore, it can be concluded that Good Treads tires last longer than Speed Tracks tires.

Answers

Good Treads' claim is false; therefore, it can be concluded that Speed Tracks tires last longer than Good Treads tires.

The car company's report provides evidence that Speed Tracks tires have a longer average lifetime compared to Good Treads tires. Therefore, we can conclude that Good Treads' claim that their tires last longer is false.

Based on the car company's report, we can conclude that Good Treads' claim that their tires last longer than Speed Tracks' tires is false. The average lifetime of a Good Treads tire is reported to be 55,000 miles, while the average lifetime of a Speed Tracks tire is reported to be 65,000 miles.

To reach this conclusion, we compare the average lifetimes of the two tire brands. Since the car company tested 20 sets of each brand's tires and measured the tread on each tire after simulating 1,000 miles of driving conditions, we can assume that these results are representative.

By comparing the average lifetimes, we see that the average lifetime of Speed Tracks tires is greater than the average lifetime of Good Treads tires. Therefore, we can conclude that Speed Tracks tires last longer than Good Treads tires, contradicting Good Treads' claim.

In conclusion, the car company's report provides evidence that Speed Tracks tires have a longer average lifetime compared to Good Treads tires. Therefore, we can conclude that Good Treads' claim that their tires last longer is false.

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Maggie ebought 52 ounces of swordfish selling for $6.92 per pund. What was the total cost?

Answers

The total cost was $22.49

What's a parabola? What's a hyperbola?

Answers

Parabola: is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape.

Hyperbola:
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows

Hope this Helps

Both parabolas and hyperbolas are an open curve which means that the arms or branches of the curves continue to infinity; they are not closed curves like a circle or an ellipse.

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