domain for f(x) = -0.25(x − 4) 2 − 6

We first compute the partial derivatives of f(x, y):

f(x, y) = 5x² + 5y² - 10x + 21

∂f/∂x = 10x - 10

∂f/∂y = 10y

The critical points of f occur where both ∂f/∂x and ∂f/∂y are equal to zero. This only happens at the point (x, y) = (1, 0). (And this point does lie inside R.)

Next, compute the Hessian matrix H(x, y) for f :

[tex]H(x, y) = \begin{bmatrix}\frac{\partial^2f}{\partial x^2} & \frac{\partial^2f}{\partial x\partial y} \\ \frac{\partial^2 f}{\partial y\partial x} & \frac{\partial^2f}{\partial y^2}\end{bmatrix} = \begin{bmatrix}10&0\\0&10\end{bmatrix}[/tex]

Since det(H(x,y)) = 100 is positive for all x and y, this means the critical point (1, 0) is a local minimum, and we have f(1, 0) = 16.

Next, we check for extrema along the boundary of R, which is comprised of a semicircle with radius 2 and the line segment connecting (-2, 0) and (2, 0).

• Parameterize the semicircular portion by

x(t) = 2 cos(t)

y(t) = 2 sin(t)

with 0 ≤ t ≤ π. Then

f(x(t), y(t)) = 20 sin²(t) + 20 cos²(t) - 20 cos(t) + 21

which is simplifies to a function of one variable t,

g(t) = 41 - 20 cos(t)

Find the extrema of g over the interval [0, π] : we have critical points when

g'(t) = 20 sin(t) = 0

which happens at t = 0 and t = π. At these points, we get local extreme values  of

•• t = 0   =>   x = 2 and y = 0   =>   f(2, 0) = 21

•• t = π   =>   x = -2 and y = 0   =>   f(-2, 0) = 61

• Over the line-segment portion, we take y = 0, so f(x, y) again reduces to a function of one variable:

f(x, 0) = 5x² - 10x + 21

Completing the square, we have

5x² - 10x + 21 = 5 (x - 1)² + 16

which has a maximum value of 16 when x = 1 (and this happens to be the critical point (1, 0) we found earlier).

So, over the region R, f(x, y) has

• an absolute maximum of 61 at the point (-2, 0), and

• an absolute minimum of 16 at (1, 0)

9514 1404 393

Answer:

  a. Z: 30,000; T: 50,000

  b. Z: 37.5%; T: 62.5%

Step-by-step explanation:

a. The "ratio units" add up to 3+5 = 8, so we can find the investment amounts by multiplying the ratio by 10,000

  Z : T = 3 : 5 = 30,000 : 50,000

__

b. Zoe's investment amount is 3/8 of the total, or ...

  3/8 × 100% = 37.5%

Toria's percentage is the rest of it:

  100% -37.5% = 62.5%

  Zoe invested 37.5% of the total; Toria invested 62.5% of the total.

Answer:

5:1

Step-by-step explanation:

5/6 won

1/6 lost

5:1=6

Answer:

5:1

Step-by-step explanation:

Total numbers of match = 6

No. of matches Peter wins = 5

Ratio of wins to lose = 5:(6-5) = 5:1

Please, don't write the answer as 1:5. Because, 5:1 and !:5 both are different so don't be confused.

O NEGATIVE ONE-HALF ........

Answer:

answer is D

Step-by-step explanation:

the answer is 1.75

============================================================

Explanation:

The order of a matrix is a way of describing its size. Specifically, the order is represented in the form [tex]m \times n[/tex], where,

The rows are laid out horizontally, and the columns are vertical.

In this case, we have 5 rows and 3 columns. Therefore, the answer is 5 x 3. The order is important because a 5 x 3 matrix is different from a 3 x 5 matrix.

Answer:

a) 5.25m^2

b) 1 can

Step-by-step explanation:

a)

area of a triangle = bh/2

=(4.2 x 4.5)/2

=18.9/2

=9.45m^2

area of a rectangle = 2(l+w)

=2 x (0.6 + 1.5)

=4.2m^2

9.45-4.2 = 5.25m^2

b) 1 can, you can't have a partial can of paint

The value of "s" as a function of other  variables is s =  (w - rh)/c

If you earn both hourly wages and a commission at your job, your gross wages, w, can be calculated with the following equation:

cs + rh = w

We will make "s" the subject of the formula:

Subtract rh from both sides

cs + rh - rh = w - rh

cs = w - rh

Divide both sides by c

cs/c = (w - rh)/c

s =  (w - rh)/c

Hence the value of "s" as a function of other  variables is s =  (w - rh)/c

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Answer:

subtract 22 from both sides then multiply 3 to both sides

Step-by-step explanation:

F/3+22=17

-22. -22

F/3 = -5

x3. x3

F= -15

Answer:

{ - 5, 19, 31 }

Step-by-step explanation:

Substitute the values from the domain ( input) into f(x) to obtain range (output )

f(- 3) = 2(- 3) + 1 = - 6 + 1 = - 5

f(9) = 2(9) + 1 = 18 + 1 = 19

f(15) = 2(15) + 1 = 30 + 1 = 31

range is { - 5, 19, 31 }

Answer: 4928

Step-by-step explanation:

Area= LxW

88 x 56= 4928

60

Step-by-step explanation:

(-47)-(-107)

(-47)-1(-107)

-47+107

=60

Size of final payment is $6392.43

Given that formula for compound interest is P(1+r/n[tex])^{nt}[/tex]

where P is initial principal

r is interest rate

n is compound times per period

t is time in years

We find all the variables in the formula in order to get our answer.

Initial principal = 96000 less 25%= 72000

interest rate = 9%

number of compound times= 12

time in years =4.25 years (whole period is 4.5 years since number of payments to complete balance will take 54 months)

Hence principal and CI after 4 years and 3 months(the second to last payment)

72000(1+0.09/12)^12x4.25=$105,393.6

We use $105,393.6 as our principal to calculate the whole interest and principal for the period in order to find last payment

105393.6(1+0.09/12)^12x0.25=$107,786.03

interest for the final month = $107786.03-$105,393.6=$2392.43

To get our final payment, we add the principal payment of $4000 to final compounded interest =$6392.43

Answer:

Step-by-step explanation:

(-7 - k)/(6 + 4) = -15

(-7 - k)/10 = -15

-7 -k = -150

-k = -143

k = 143

Answer:

1. tenths = 0.3

   thousandths = 0.346

   hundred thousandths = 0.34600

   hundredths = 0.35

2. thousandths = 20.006

    hundredths = 20.01

   hundred thousandths = 20.00642

Step-by-step explanation:

Hope it helps

Answer:

Area = 100cm²

Perimeter = 48cm

Answer:

[tex]solution 1: \ \frac{7+\sqrt{65}}{2};[/tex]

[tex]solution2: \ \frac{7-\sqrt{65} }{2}.[/tex]

Step-by-step explanation:

x²-7x-4=0;

D=[b²-4ac]=49+16=65.

[tex]\left[\begin{array}{ccc}x=\frac{7+\sqrt{65}}{2} \\x=\frac{7-\sqrt{65}}{2} \end{array}[/tex]

Using a system of equations, the height of the tree would be 12.80 meters.

Equating the height and shadows of the tree and pole :

Height of pole / shadow of pole = height of tree / shadow of tree

Representing the equation thus :

Let the height of the tree = h

3/4 = h/17

Cross multiply

4h = 17 × 3

4h = 51

Divide both sides by 4

h = 51/4

h = 12.75.

Hence, the height of the tree to the nearest tenth of a meter is 12.80 meters.

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Answer:

no entiendo, perdóname, suerte

Answer:

Step-by-step explanation:

Given:

Simplify equation for perimeter and solve for W:

Find L:

Answer:

2/3 ​sandwich.

Step-by-step explanation:

No Explanation

Answer:

5

Step-by-step explanation:

The answer is 60/12 becuase we can simfly both of theese mixed numbers into an improper fraction, wich will be 4/3*15/4. We do the multiplication, and we get 60/12. 60/12 can be simplifed to 5 becuase 60 divided by 12 is 5

Answer:

this is the correct answer happy

Answer:

C, 18.

Step-by-step explanation:

Since 24 divided by 8 is 3, which means that

8x3 = 24, then the amount of boys must be solved using the equation 6x3 which equals 18.

The equation that can be used to determine the original length of the pool is 205.85 = 11.6(l + 3.7)

Let l be the original length of the pool.

Since the length of the pool is increased by 3.7 m, the new length L = l + 3.7.

Also, the area of the pool A = 205.85 square meter and it is a rectangle with width, W = 11.6 meters,

So, A = LW

Substituting the values of the variables into the equation, we have

A = LW

205.85 = (l + 3.7)11.6

So, the equation that can be used to determine the original length of the pool is 205.85 = 11.6(l + 3.7)

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Answer:

Step-by-step explanation

205.85=11.6(l+3.7)

Answer:

Step-by-step explanation:

Answer:

Mary:604

Mona:796

Step-by-step explanation:

Answer:

Ms. Mary spend $302

Ms. Mona spend $ 686

Step-by-step explanation: We know that Ms. Mary spend $192 less than Ms. Mona.

So we are to divide $988 by 2 which is 494

now both have $494 each but Ms. Mona spend 192 so we are to add 192 + 494 to get the right sum of how much money Ms. Mona have and take away 494-192 to get the right sum of how much money Ms. Mary spend.

192 + 494 = 686

494 - 192 = 302

Answer:

Step-by-step explanation:

Solve for k.

14k+3k–16k=5

17k - 16k = 5

k = 5

500 children ages 5-11 were asked the following question:

"Do you read books for fun every day?"

Results are shown in the two-way table below.

             Boy    Girl     Total

Yes         48      75       123

No          182     195     377

Total      230    270     500

A proportion is the number of desired outcomes divided by the number of total outcomes.

There are 500 children and 230 are boys.

230/500 = 0.46

There are 230 boys and 48 read books for fun every day.

48/230 = 0.21

There are 270 girls and 75 read books for fun every day.

75/270 = 0.28

Based on the results from b and c, it would seem more likely for a girl to read books for fun than for a boy.

There are 500 children and 123 read books for fun.

P = 123/500 = 0.25

There are 500 children and 75 are girls who read books for fun.

P = 75/500 = 0.15

There are 500 children. The favorable cases are the girls (270) plus the children who read books for fun (123) minus the girls who read books for fun (75).

P = (270 + 123 - 75)/500 = 0.64

There are 270 girls and 75 of them read books for fun.

P = 75/270 = 0.28

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