Find the extreme values of f subject to both constraints f(x,y) = 2x^2+3y^2-4x-5, x^2 + y^2 <=16

Answers

Answer 1
[tex]f(x,y)=2x^2+3y^2-4x-5[/tex]
[tex]f_x=4x-4=0\implies x=1[/tex]
[tex]f_y=6y=0\implies y=0[/tex]

[tex]f(x,y)[/tex] has only one critical point at [tex](x,y)=(1,0)[/tex]. The function has Hessian

[tex]\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}4&0\\0&6\end{bmatrix}[/tex]

which is positive definite for all [tex](x,y)[/tex], which means [tex]f(x,y)[/tex] attains a minimum at the critical point with a value of [tex]f(1,0)=-7[/tex].

To find the extrema (if any) along the boundary, parameterize it by [tex]x=4\cos t[/tex] and [tex]y=4\sin t[/tex], with [tex]0\le t<2\pi[/tex]. On the boundary, we have

[tex]f(x(t),y(t))=F(t)=2(4\cos t)^2+3(4\sin t)^2-4(4\cos t)-5=32\cos^2t+48\sin^2t-16\cos t-5[/tex]
[tex]F(t)=35-16\cos t-8\cos2t[/tex]

Find the critical points along the boundary:

[tex]F'(t)=16\sin t+16\sin2t=16\sin t+32\sin t\cos t=16\sin t(1+2\cos t)=0[/tex]
[tex]\implies t=0,\dfrac{2\pi}3,\pi,\dfrac{4\pi}3[/tex]

Respectively, plugging these values into [tex]F(t)[/tex] gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when [tex]F(t)=47[/tex].

Now, solve for [tex]x,y[/tex] for both cases:

[tex]t=\dfrac{2\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=2\sqrt3\end{cases}[/tex]

[tex]t=\dfrac{4\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=-2\sqrt3\end{cases}[/tex]

so [tex]f(x,y)[/tex] has two absolute maxima at [tex](x,y)=(-2,\pm2\sqrt3)[/tex] with the same value of 47.

Related Questions

What is 3x5x6x19x500

Answers

3x5x6x19x500 is 855,000
The answer is 855,000. Hope this helps :)

The point-slope form of the equation of the line that passes through (–4, –3) and (12, 1) is y – 1 = (x – 12). What is the standard form of the equation for this line?

Answers

y= x-11   bc...
y-1=x-12
 +1   +1
y=x-11

It takes one machine 25 minutes to pick a bale of cotton and another machine 30 minutes to compete the same job. If both machines are used, how long will it take them to pick a bale of cotton?

Answers

Answer:

13.64 minutes

Step-by-step explanation:

The first machine works at the rate x = 1 bale/ 25 min.

The second machine works at the rate y = 1 bale/30 min.

The combined rate is

x + y = 1/25 + 1/30 bales/min

        = (30 + 25)/(25*30) = 55/750

        = 11/150 bales/min

The time for both machines to pick 1 bale of cotton is

(1 bale)/(11/150 bales/min) = 150/11 min = 13.64 min

So if you take rsm, you have to input 13.64 otherwise it won't work. :)

What is the GCF of 12,28,36?

Answers

to find the GCF, you have to list all factors of each number and find the first common one.

12 = 1, 2, 3, 4, 6, 12

28 = 1, 2, 4, 7, 14, 28

36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

the GCF of numbers 12, 28, 36 is 4.

hope this helps, God bless!

What was the total production of gasoline and kerosene combined?

Answers

それでも、日本人の食生活も急速に変化してきています。

One tank is filling at the rate of 3/4 gallon / 2/3 minute a second tank is filling at the rate of 3/8 gallon per 1/2 minute which tank is filling faster equation

Answers

[tex]\bf \cfrac{\quad \stackrel{gallons}{\frac{3}{4}}\quad }{\stackrel{minutes}{\frac{2}{3}}}\implies \cfrac{3}{4}\cdot \cfrac{3}{2}\implies \cfrac{9}{8}~g/m \\\\\\ \cfrac{\quad \stackrel{gallons}{\frac{3}{8}}\quad }{\stackrel{minutes}{\frac{1}{2}}}\implies \cfrac{3}{8}\cdot \cfrac{2}{1}\implies \cfrac{3}{4}~g/m\\\\ -------------------------------[/tex]

[tex]\bf \cfrac{3}{4}\cdot \cfrac{2}{2}\implies \cfrac{6}{8}\quad \textit{clearly then }\cfrac{9}{8}\textit{ is larger than }\cfrac{6}{8} \\\\\\ \stackrel{\stackrel{tank's~volume}{gallons}}{y}=\cfrac{9}{8}\stackrel{minutes}{x}[/tex]

The surface area of a cube varies directly as the square of the length of an edge. true or false

Answers

The surface area of a cube is 6 times the surface area of one face.

If the side of the cube is x, then the surface area of one side is x^2, and the total surface area is 6x^2 (because there are 6 squares making up the surface of the cube).

Here, we have A(x) = 6 x^2, which DOES vary directly as the square of the length of an edge, denoted by x.

First right answer gets brainliest

Answers

Problem:
find line perpendicular to 4x+7y+3=0 passing through (-2,1).

Solution:
We will first find the general form of the perpendicular line.
The perpendicular line has the form
7x-4y+k=0 .........................(1)
by switching the coefficients of x and y, and switching the sign of one of the two coefficients.  This way, the slope of (1) multiplied by that of the original equation is -1, a condition that the two lines are perpendicular. The value of k is to be determined from the given point (-2,1).

To find k, we substitute x=-2, y=1 into equation (1) and solve for k.
7(-2)-4(1)+k=0
=>
k=14+4=18
Therefore the required line is
7x-4y+18=0

If you know the slope of a line and a point the line passes through, you can find the equation of the line using the formula:

y - y_1 = m(x - x_1)

You are given a point the line passes through, (-2, 1).
Now you need to slope of your line.
You are given the equation of a line that your line is perpendicular to.
The slopes of perpendicular lines are negative reciprocals.
If you find the slope of the given line, you can find the slope of the perpendicular line.

Given line:

4x + 7y + 3 = 0

Solve for y:

7y = -4x - 3

y = (-4/7)x - 3/7

The slope of the given line is -4/7.
The slope of the perpendicular is 7/4.

Now we can find the equation of the line using the given point and the slope we just found.

y - y_1 = m(x - x_1)

y - 1 = (7/4)(x - (-2))

y - 1 = (7/4)(x + 2)

4y - 4 = 7x + 14

7x - 4y + 18 = 0

answer plz got cunfused on question and cant solve

Answers

area of parallogram now = 8*10 = 80 square units

 if the sides are multiplied by 4 the new dimensions would be

8*4 = 32, 10*4 = 40

 area would be 32 *40 = 1280 square units

1280 /80 = 16

 the area multiplied 16 times which is the multiple squared ( 4^2 = 16)

Which figure represents an undefined term?

Answers

In geometry, an undefined term is a basic figure that cannot be defined in terms of other figures.
The undefined terms in geometry are: point, line and plane.

Since you have not attached any figures, I cannot tell you which one is correct. Just choose the figure that is either a point, a line or a plane.

Why do some absolute value equations have no solution?

Answers

Because both positive and negative values have a positive absolute value.
An absolute value is the distance from 0 on a number line so it must be a positive number, and so  because of that it is impossible to get or equal -1, this equation has no solution.

Ignatz repeatedly rolls a fair $6$-sided die. What is the probability that he rolls his first $5$ before he rolls his second (not necessarily distinct) even number?

Answers

The probability that Ignatz rolls his first 5 before he rolls his second (not necessarily distinct) even number is: 13/32

Events:

Define Event A: Rolling 5 before an even number.

Define Event B: Rolling an even number before a 5 (complementary event to event A).

Sample Space: - The total number of outcomes for rolling a die 5 times is [tex]6^5[/tex]  (each roll has 6 possibilities).Favorable Outcomes for Event A:- We can achieve event A in various scenarios where Ignatz rolls a 5 before encountering an even number. Let's consider each case:

Case 1: 5 in the first roll (1 way). Then, any other number (not 2, 4, or 6) in the remaining rolls (5 possibilities each).

Favorable outcomes: [tex]1 * 5 * 5 * 5 * 5.[/tex]

Case 2: 5 in the second roll (similar to case 1).

Favorable outcomes: [tex]5 * 1 * 5 * 5 * 5.[/tex]

We can continue using the same logic for cases where 5 appears in the third, fourth, or fifth roll.

Total Favorable Outcomes for Event A:

Since there are multiple ways to achieve event A, we add the favorable outcomes from each case.Total Favorable Outcomes (A) = Favorable Outcomes (Case 1) + Favorable Outcomes (Case 2) + ... + Favorable Outcomes (Case 5)

= [tex]1 * 5^4 + 5 * 1 * 5^4 + 5^2 * 1 * 5^3 + 5^3 * 1 * 5^2 + 5^4 * 1 * 5[/tex]

Probability of Event A:

Probability (A) = Favorable Outcomes (A) / Total Outcomes

[tex]= ( 1 * 5^4 + 5 * 1 * 5^4 + 5^2 * 1 * 5^3 + 5^3 * 1 * 5^2 + 5^4 * 1 * 5) / 6^5[/tex]

Probability of Event B (complementary event):

Event B is the opposite of event A. Since events A and B cover all possibilities, their probabilities sum to 1.

Probability (B) = 1 - Probability (A)

Therefore, the probability of rolling a 5 before an even number (event A) is 1 - Probability (B).

(01.03) A box has dimensions of 14 inches long, 1.5 feet wide, and 7 inches high. What is the volume of the box

Answers

1.5 feet = 12 * 1.5 = 18 inches

14 x 18 x 7 = 1764 cubic inches

Help please!!! practical math

Answers

This is a problem that would take a very long time to do by hand. So a calculator is recommended. If you have a TI calculator (such as the TI 83) then use that. The calculator I'm going to use is GeoGebra. I've attached the results in the image attachment.

The line of best fit (aka the regression line) that GeoGebra produces is the equation y = 1.862 - 0.138 

Which means the answer is choice A

Write the equation in spherical coordinates. (a) 4z2 = 5x2 + 5y2 (b) x2 + 3z2 = 4

Answers

Refer to the diagram shown below.

The transformation from the Cartesian coordinate system (x,y,z) to the spherical coordinate system (r,θ,φ) is given by
x = rsinφcosθ
y = rsinφsinθ
z = rcosφ

Part a.
The given equation is 4z² = 5x² + 5y².
In spherical coordinates, obtain
4r²cos²φ = 5r²sin²φ cos²θ + 5r²sin²φ sin²θ
             = 5r²sin²φ(cos²θ + sin²θ)
             = 5r²sin²φ
4/5 = tan²φ
tanφ = 2/√5 = (2/5)√5

Answer:  tanφ = (2/5)√5

Part b.
The given equation is x² + 3z² = 4
In spherical coordinates, obtain
r²sin²φ cos²θ + 3r²cos²φ = 4

Answer:  r²(sin²φ cos²θ + 3cos²φ) = 4

Final answer:

In spherical coordinates, equations in the form of [tex]4z^2 = 5x^2 + 5y^2[/tex] can be written as [tex]4ρ^2cos^2(θ) = 5ρ^2sin^2(θ)[/tex], and equations in the form of x^2 + 3z^2 = 4 can be written as ρ^2sin^2(θ) + 3ρ^2cos^2(θ) = 4.

Explanation:

(a) In spherical coordinates, the equation [tex]4z2 = 5x2 + 5y2[/tex] can be written as [tex]4ρ2cos2(θ) = 5ρ2sin2(θ)[/tex]. Here, ρ represents the distance from the origin to the point and θ represents the angle measured from the positive x-axis to the point in the xy-plane.

(b) For the equation x2 + 3z2 = 4, we can rewrite it in spherical coordinates as[tex]ρ2sin2(θ) + 3ρ2cos2(θ) = 4.[/tex]

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Name the triangle with the following characteristics. sides: 5 cm, 6 cm, 7 cm; Angles: 75° and 60°.

Answers

If the angles are below 90 degrees this would make this triangle an acute triangle 

The answer is: The triangle is a SCALENE Triangle.

As all the sides of the triangle are unequal. Therefore,it is a scalene triangle.

What are the 3 main types of triangles?

According to the length of sides, triangles can be classified into 3 categories i.e., Scalene, Isosceles, and Equilateral triangles.

A scalene triangle is a triangle in which all three sides are of different lengths, and all three angles are of different measures. However, the sum of all the interior angles is always equal to 180 degrees.

An isosceles triangle is a triangle in which two sides have the same length.

An equilateral triangle is a triangle with all three sides of equal length, corresponding to what could also be known as a "regular" triangle.

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Three friends all live on the same street that runs west to east. Ben lives 4 blocks from Ann. Carl lives 2 blocks from Ben. If the street is represented by a number line and Ann​'s house is located at​ 0, what are the possible locations for Carl​'s ​house?

Answers

Answer:

The possible locations are,

2, -2, 6 or -6

Step-by-step explanation:

Given,

The location of Ann's house in the number line = 0,

∵ Ben lives 4 blocks from Ann.

So, the possible location of Ben's house =  4 ( right of ann's house ) or - 4 ( left of ann's house )

Now, Carl lives 2 blocks from Ben,

So, the possible location of Carl's house

Case 1 : If Ben's house location is 4 = 2 ( left of ben's house ) or 6 ( right of ben's house )

Case 2 : If Ben's house location is -4 = -2 ( right of ben's house ) or -6 ( left of ben's house )

The possible locations of Carl's house are - 6, 6, 2 or - 2

Let :

Ben's location = B

Carl's position = C

Ann's position = A

Using the Number line illustration :

Ann's position = 0

Ben's possible positions :

4 blocks(units) from Ann = ± 4

Carl's location :

2 blocks from Ben-4 ± 2 = (-4 - 2) or (-4 + 2) = - 6 or - 2 +4 ± 2 = (4 + 2) or (4 - 2) = 6 or 2

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You want to find out how many baseball cards Ryan had to begin with. Which situation about his card collection best represents the equation
x/2-1=24

A. Half of his cards were ruined after they got wet, and then he lost 1 card. After that, he had 24 cards left.

B. He lost 1 card, and then half of his cards were ruined after they got wet. After that, he had 24 cards left.

C. Half of his cards were ruined after they got wet, and then he bought 1 more card. After that, he had 24 cards left.

D. He bought 1 card, and then half of his cards were ruined after they got wet. After that, he had 24 cards left.

Which situation about Connor’s commission best represents the inequality 40 + 10x ≥ 100?



A.Connor earns $10 per day plus a $40 commission on each item he sells. How much will he earn if he sells 100 items?

B.Connor earns $40 per day plus a $10 commission on each item he sells. How much will he earn if he sells 100 items?

C.Connor earns $40 per day plus a $10 commission on each item he sells. How many items must he sell to earn at least $100?

D.Connor earns $40 per day plus a $10 commission on each item he sells. How many items must he sell to earn no more than $100?

Answers

The first one is a. If you solve for x, equals 50. And if you retrace the steps described in a, describes the situation. Lost half, then one more.

The second is c, the inequality describes the condition and solves how many commissions he needs to earn at least 100 or more dollars a day.

A hardware store has 147 containers of paint. If each container holds 5 gallon of paint, how many gallon of paint are at the store?

A.235 B.595 C.735 D.905

Dale chose A as the correct answer how did he get that answer?

I know the answer is C but can't explain why he chose A

Answers

The answer is 147 containers * 5 gallons/container = 
735 gallons of paint

Dale forgot the 1 in 147, so he multiplied 47 x 5  = 235 

A bird leaves its nest and travels 18 miles per hour downwind for x hours. On the return trip, the bird travels 3 miles per hour slower and has 5 miles left after x hours.

Answers

d = distance down wind d - 5 = distance back time = distance ÷ speed x = d/18 x = (d - 5)/(15) 
so d/18 = (d - 5)/15. Cross multiply 18d - 120 = 15d = 40-120 = -5d 24 = d so the whole trip assuming the bird gets back to the starting point is 48 miles..

What is the greatest common factor of 12 and 42

Answers

The greatest common factor for 12 and 42 is 6

The greatest common factor of 12 and 42 is 6.

What is Number system?

A number system is defined as a system of writing to express numbers.

We need to find the greatest common factor of 12 and 42.

The greatest common factor (GCF) of a set of numbers is the largest factor that all the numbers share.

The factors of 12 are  1, 2, 3, 4,6 and 12.

The factor of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.

Let us find the common factors of 12 and 42.

The common factors are 1, 2 , 3, 6.

Among the common factors which is greatest value.

Among 1,2,3 and 6 the number 6 is greatest.

Hence,  the greatest common factor of 12 and 42 is 6.

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Someone please help me with # 10

Answers

where is the qustion


Solve the formula M = 2P + 3Q for the variable Q.

Answers

To solve for the formula Q, simply rearrange the variables and factor the appropriate number.

M = 2P + 3Q

M - 2P = 3Q

(M - 2P)/3 = Q.

Last year Jeremy had $7900 in medical expenses. If his AGI was $89,000, will he be able to deduct his medical expenses and if so, what amount?

Answers

medical expenses are deducted on schedule A - which is required to be over 7.5% of AGI

Answer:

He will be able to deduct his medical expenses by $1225.

Step-by-step explanation:

If his medical expenses are over 7.5% of his AGI he can reduce it to 7.5%.

His AGI was $89,000. How much is 7.5% of that?

0.075*89000 = $6675.

His expenses are over 7.5%, so yes, he can reduce.

He can reduce from $7900 to $6675, so 7900 - 6675 = $1225

Paula has a digas that weighs 3 times as much as Carla's. The total weight of the dogs is 48 lb how much does Paula dog weigh. Draw a diagramar ti find the weighs of Pailas dog

Answers

the answer would be 48 divided by 3
so the answer would be 16
the answer is 16 because when u division

ΔABC with vertices A(-3, 0), B(-2, 3), C(-1, 1) is rotated 180° clockwise about the origin. It is then reflected across the line y = -x. What are the coordinates of the vertices of the image?

A(. A'(0, 3), B'(2, 3), C'(1, 1)
B(. A'(0, -3), B'(3, -2), C'(1, -1)
C(. A'(-3, 0), B'(-3, 2), C'(-1, 1)
D(. A'(0, -3), B'(-2, -3), C'(-1, -1)

Answers

Answer: B

Picture 1 shows the reflection of ABC 180° clockwise about the origin. To draw A'B'C' we draw the line segments AO, BO, and CO and extend them such that O is the midpoint of line segments AA', BB', and CC' respectively.

Picture 2 shows the line y=-x (the yellow line.) This is the line containing all points of the form (x, -x), that is, this points contains (1, -1), (-2, 2) etc...

We see that point C' is already on this line, and we draw the reflections of points B and C with respect to this line, to find B'' and C'' with coordinates as shown in the figure.



Answer:

B (. A'(0, -3), B'(3, -2), C'(1, -1)

Step-by-step explanation:

Two different ways to find the difference for 12-3

Answers

You can either ask "How much is left when you take 3 away from 12?", or you can ask "How far apart are the numbers 12 and 3 on the number line?"

Another way you could think about it is "what number do I have to add to 3 to get to 12?"
First, you could count from 3 up to 12 and record the number between them. Second, you could use mental math and try to figure out mentally. The answer is 12-3= 9

How many pounds of cashew nuts worth 75 cents a pound must be mixed with 10 pounds of pecans worth $1.50 a pound to make a mixture worth $1.00 a pound? Which of the equations could be used to solve for p, the pounds of cashew nuts?

Answers

Let c represent the weight of cashews and p the weight of pecans.

Then c + 10 = total weight of the nut mixture.

An equation for the value of the mixture follows:

 $1.50(10 lb) + $0.75c = (c+10)($1.00)

Solve this equation for c:  15 + .75c = c + 10.  Subtract .75c from both sides:

15 = 1c - 0.75c + 10.  Then 5=0.25c, and c = 5/0.25, or 20.

Need 20 lb of cashews.


Check:  the pecans weigh 10 lb and are worth $1.50 per lb, so the total value of the pecans is $15.  The total value of the cashews is (20 lb)($0.75/lb), or $15.     Does (20 lb + 10 lb)($1/lb) = $15 + $15?  Yes.  So c= 20 lb is correct.

what is the equation of the perpendicular to y=2/3x+1 that passes through the point (12,-6)

Answers

well, first off let's check what is the slope of y = 2/3x+1.... .hmmmmm, notice    [tex]\bf y=\stackrel{slope}{\cfrac{2}{3}}x+1[/tex]  , since the equation is already in slope-intercept form, we can see it has a slope of 2/3.

well then, a line perpendicular to that one, will have a negative reciprocal to it.

[tex]\bf \textit{perpendicular, negative-reciprocal slope for slope}\quad \cfrac{2}{3}\\\\ slope=\cfrac{2}{{{ 3}}}\qquad negative\implies -\cfrac{2}{{{ 3}}}\qquad reciprocal\implies - \cfrac{{{ 3}}}{2}[/tex]

so, we're really looking for the equation of a line whose slope is -3/2 and goes through 12, -6.

 [tex]\bf \begin{array}{lllll} &x_1&y_1\\ % (a,b) &({{ 12}}\quad ,&{{ -6}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies -\cfrac{3}{2} \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})}\implies y-(-6)=-\cfrac{3}{2}(x-12) \\\\\\ y+6=-\cfrac{3}{2}x+18\implies y=-\cfrac{3}{2}x+12[/tex]

How do the 3's compare in 350 and 403?

Answers

In 350, the 3 is in the hundred's place

in 403, the 3 is in the one's place

hope this helps
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