Find the absolute maximum and absolute minimum of the function
f(x, y) = y^2 — x^2 + 4xy
over the region R that is the square with vertices (−1,0); (0, 1); (1,0) and (0, –1).
Answers
The absolute maximum and absolute minimum of the given function over the region R that is the square with vertices (−1,0); (0, 1); (1,0) and (0, –1) are 1 and -1, respectively.
To find the function's absolute maximum and absolute minimum, f(x, y) = y^2 — x^2 + 4xy, we need to determine the critical points in the given square region R and then use the Second Derivative Test to classify them.
Then we must evaluate the function at each vertex of R and select the greatest and smallest values as the absolute maximum and minimum values of f(x, y), respectively. So let's calculate the critical points of the given function:
∂f/∂x = -2x + 4y = 0 ...............(1)
∂f/∂y = 2y + 4x = 0 ................(2)
From (1) and (2),
we have x = 2y and y = -2x/4
⇒ y = -x/2.
Substituting this value of y in equation (1), we get x = -y.t
Now, we can write the point (x, y) = (-y, -x/2) as the critical point.
To classify these critical points as maximum, minimum or saddle point,
we can write the Second Derivative Test.
D(f(x, y)) = ∂²f/∂x² ∂²f/∂x∂y∂²f/∂y∂x ∂²f/∂y²
= (-2) (4) (4) (-2) - (4)²
= -16 < 0
Thus, we have a saddle point at (-y, -x/2). The greatest and smallest values are the absolute maximum and minimum values of f(x, y), respectively. Thus, we concluded that the absolute maximum and absolute minimum of the given function over the region R that is, the square with vertices (−1,0); (0, 1); (1,0) and (0, –1) are 1 and -1, respectively.
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Related Questions
f(x) = 4x2 – 10x + 2
Find f(-7)
Answers
Answer:
\(f( - 7) = 80\)
Step-by-step explanation:
\(x = - 7 \\ 4 \times 2 - 10 \times - 7 + 2 \\ = 8 + 70 + 2 \\ = 80 \\ \\ hope \: it \: helps \\ pls \: mark \: it \: as \: brailiest....\)
Use the model to find 2/4 x 16
Answers
Answer:
8
Step-by-step explanation:
Reduce the fraction to 1/2 then multiply it by 8 which is 8:) this this helps! Plzz mark brainliest
Answer:
the awnser is 8
Step-by-step explanation:
1.divide the numbers
2/4 * 16
1/2 * 16
2. multiply the numbers
1/2 x 16
=8
hope this helps man
If A1="C", what will the formula =IF(A1="A",1,IF(A1="B",2,IF(A1= " D=,4,5))) return?
5
3
4
2
Answers
The formula will return 5, because none of the conditions in the nested IF statement are true for the value of A1 being "C".
The formula =IF(A1="A",1,IF(A1="B",2,IF(A1="D",4,5))) is a nested IF statement that checks the value of cell A1 and returns a corresponding value based on the conditions.
In this case, the value of A1 is "C". Therefore, the first condition, A1="A", is not true, so the formula moves on to the second condition, A1="B". This condition is also not true, so the formula moves on to the third condition, A1="D". However, this condition is also not true, because the third condition has a typo, where there is an extra space before the "D". Therefore, the formula evaluates the final "else" option, which is 5.
Thus, the formula will return 5, because none of the conditions in the nested IF statement are true for the value of A1 being "C".
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In Fechner's Law as one variable increases arithmetically, the other variable increases ____.
a. arithmetically
b. geometrically
c. metaphysically
d. mathematically
e. psychophysically
Answers
In Fechner's Law, as one variable increases arithmetically, the other variable increases geometrically.
In Fechner's Law, as one variable increases arithmetically, the other variable increases: b. geometrically.
Fechner found in his research that different people have different sensitivities to certain stimuli. For example, the ability to see differences in light can affect the quality of a person's vision. It has also been noted that people are sensitive to changes in stimuli due to emotional impact. He used this to create another version of Weber's law, which he called die Maß formel, or "measurement formula." Fechner's law states that the force of thought is proportional to the logarithm of the force.
If the intensity of the stimulus is tripled (eg 3×1), the sensitivity doubles from its original value (eg 1+1).
If the intensity of the stimulus is tripled again (eg 3×3×1), the corresponding sensitivity will be three times its original value (eg 1+1+1).
So, for the stimulus intensity to double, the perceived intensity only increases.
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Evaluating functions
Answers
Step-by-step explanation:
Hope this helps please like and mark as brainliest
Same question I answered it but I don’t know if it’s correct
Answers
Answer:
Yes it is correct
Step-by-step explanation:
A good way to know it's correct is the "three times a number" because that's how you would know it's 3x
Help me pls.........
Answers
Answer:
The account balance will be $2,614.16 after 10 years ⇒ D
Step-by-step explanation:
The formula for compound interest, including principal sum is:
\(A=P(1+\frac{r}{n})^{nt}\), where
A is the future value of the investment/loan, including interestP is the principal investment amount r is the annual interest rate (decimal)n is the number of times that interest is compounded per unit tt is the time the money is invested or borrowed forLet us use this rule to solve the question
∵ Ted invests $1,077 in a savings account with a fixed annual interest
rat 9% compounded three times per year
∴ P = 1,077
∴ r = 9% = \(\frac{9}{100}\) = 0.09
∴ n = 3 ⇒ three times per year
∵ The time is ten years
∴ t = 10
Substitute these values in the rule above to find A
∵ \(A=1077(1+\frac{0.09}{3})^{3(10)}\)
→ Use your calculator to find the answer
∴ A = 2,614.161681
→ Round it to the nearest cent (2d.p.)
∴ A = $2,614.16
∴ The account balance will be $2,614.16 after 10 years.
i need the answers to this
Answers
Answer:
(4,4)
Step-by-step explanation:
PLS HELP ME WITH THIS ASAPP
Answers
The value of the function f(x) at x = 1.477.
f(1.477) = 32.857
The value of the function g(x) at x = 20.
g(20) = 1.729
We have,
Two functions:
f(x) = b^x
g(x) = log_b(x)
Now,
From the table,
f(0.699) = b^0.699 = 5
Taking the logarithm of both sides with base b.
log_b(5) = 0.699
b = 5^(1/0.699)
b = 8.343
Now, we can find the value of f(1.477).
f(1.477) = b^1.477 = 8.343^1.477 = 32.857
Similarly, we can find the value of g(20).
g(20) = log_b(20) = log_8.343(20) = 1.729
Thus,
The value of the function f(x) at x = 1.477.
f(1.477) = 32.857
The value of the function g(x) at x = 20.
g(20) = 1.729
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Problem 9. (10 pts)
Let
1
A 2 2 2 2
(a) (3pts) What is the rank of this matrix?
1 2 1 1
(b) (7pts) Assuming that rank is r, write the matrix A as
A = +...+uur.
for some (not necessarily orthonormal) vectors u1,..., ur, and v1,..., Ur. Hint: Do not try to compute SVD, there is a much simpler way by observation: find a rank one matrix u that looks "close" to A and the consider A-uu.
Answers
The answer based on matrix is (a) The rank of the matrix is 2. , (b) the matrix A is = [7, 6, 1, 1].
Let
a) The rank of the matrix is 2.
b) Considering the rank as r, we can write the matrix A as A = +...+uur, for some (not necessarily orthonormal) vectors u1,..., ur, and v1,..., Ur.
We know that the rank of the given matrix is 2.
It means that there must be two independent vectors in the rows or columns of A. We observe that columns 2 and 4 of the given matrix are linearly dependent on the first two columns. Hence, we can rewrite the matrix as:
We observe that the first two columns are linearly independent, which are u1 and u2.
Using these vectors, we can write the given matrix as A = u1vT1 + u2vT2, where vT1 and vT2 are row vectors.
A rank-one matrix can be written in this form, and we know that the rank of A is 2.
This means that there must be one more vector u3, and it is orthogonal to both u1 and u2.
We can compute it using the cross product of u1 and u2.
We get:
u3 = u1 × u2 = [2, -2, 0]T
Now we can compute vT1 and vT2 by finding the null space of the matrix formed by u1, u2, and u3.
We get:
vT1 = [-1, 0, 1, 0]andvT2 = [1, 1, 0, -1]
Finally, we can write the matrix A as A = u1vT1 + u2vT2 + u3vT3, where vT3 is a row vector given by:
vT3 = [0, -1, 0, 1]
Therefore, we have: A = (1, 2, 1, 1) (-1 0 1 0) + (2, 2, 2, 2) (1, 1, 0, -1) + (2, -2, 0, 0) (0, -1, 0, 1)= [3, 0, 1, -1]+ [4, 4, 2, 2]+ [0, 2, -2, 0]
= [7, 6, 1, 1]
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Which of the following is equivalent to 8(5f – 7) + 3f
A- 40f – 7 +3f
B- 43f – 7
C- –13f
D- 43f – 56
Answers
The student council is selling school supplies as a fund raiser. Josh bought four pencils and two ballpoint pens for $0.74. Zach bought six pencils and five ballpoint pens for $1.53. Write a system of equations that can be used to find the cost of a pencil and the cost of a ballpoint pen where "p" represents pencils and "b" represents ballpoints pens.
Group of answer choices
A 4p+2b=.744 p + 2 b = .74
6p+5b=1.536 p + 5 b = 1.53
B4p-2b=.744 p − 2 b = .74
6p-5b=-1.536 p − 5 b = − 1.53
C 4b+2p=.744 b + 2 p = .74
6b+5p=1.536 b + 5 p = 1.53
D 4p+2b=1.534 p + 2 b = 1.53
6p+5b=.74
Answers
Answer:
A
4p+2b= .74
6p+5b=1.53
Which set of ordered pairs does not represent a function?
O {(4,4), (3, -8), (7,-6), (-5, -3)}
o {(-6,6), (8, 2), (8, -1), (3,3)}
o {(-4,-6), (-9, -7), (6, 7), (0, -6)}
o {(9, 5), (-5, -5), (0, 9), (8,3)}
Answers
Answer:
The second choices is the answer
{(-6,6), (8, 2), (8, -1), (3,3)} as you can see there's a repeating value of x or the Domain which is the 8.
Automatic Transmissions, Inc., has the following estimates for its new gear assembly project: price = $940 per unit; variable cost = $340 per unit; fixed costs = $3.4 million; quantity = 53,000 units. Suppose the company believes all of its estimates are accurate only to within ±15 percent. What values should the company use for the four variables given here when it performs its best-case scenario analysis? What about the worst-case scenario?
Answers
In the worst-case scenario, the company should use the following values: price = $799 per unit, variable cost = $289 per unit, fixed costs = $2.89 million, and quantity = 60,950 units.
In the best-case scenario analysis for Automatic Transmissions, Inc.'s new gear assembly project, the company assumes the upper limit of the ±15 percent range for its estimates. For the price per unit, they take a 15 percent increase, resulting in a value of $1081. Similarly, the variable cost per unit is increased by 15 percent to $391. The fixed costs are also adjusted upwards by 15 percent, reaching $3.91 million. Finally, the quantity is decreased by 15 percent, leading to a value of 45,050 units.
On the other hand, in the worst-case scenario analysis, the company assumes the lower limit of the ±15 percent range for its estimates. The price per unit is decreased by 15 percent, resulting in $799. The variable cost per unit is decreased to $289. The fixed costs are adjusted downwards to $2.89 million. Lastly, the quantity is increased by 15 percent to 60,950 units.
Therefore, in the worst-case scenario, the company should use the following values: price = $799 per unit, variable cost = $289 per unit, fixed costs = $2.89 million, and quantity = 60,950 units
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Can someone please help me, I'm stuck
Answers
what is the perimeter of a polygon with vertices at (-3,1) (5,1) (-3,4) (5,4)
Answers
The perimeter of the polygon with the vertices is: 22 units.
How to Find the Perimeter of a Polygon?To find the perimeter of a polygon, we need to add all the side lengths together.
Given the polygon has the vertices (-3, 1), (5, 1), (-3, 4), and (5,4), we need to find the distance between each vertices in order to find the perimeter of the polygon.
Let:
A = (-3, 1)
B = (5, 1)
C = (-3, 4)
D = (5,4)
The vertices have been plotted on the graph attached below. Therefore:
Perimeter of the polygon = CD + BD + AB + AC
Perimeter = 8 + 3 + 8 + 3 = 22 units.
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Plzz answer and thanks
Answers
Answer:
Erosion
Step-by-step explanation:
Erosion is the movement of sediment from one place to another. Erosion is the mechanical process of wearing or grinding somethin down. It condition in which the earth’s surface is worn away by the action of water and wind.
Therefore, the answer is erosion.
If this helps please mark as brainliest
Given: 3x < 3.
Choose the solution set.
{x | x > 0}
{x | x < 0}
{x | x > 1}
{x | x < 1}
if you know this can you help ihave to have an 85 or above in math
Answers
Answer:
Divide 3 on both sides to get x < 1
Step-by-step explanation:
Hope this helps :)
prove that for all integers a, band c, (a divides b) and (b divides c) a divides.c.
Answers
To prove that for all integers a, b, and c, (a divides b) and (b divides c) implies a divides c, we use the transitive property of divisibility.
The transitive property of divisibility states that if a number x divides a number y and a number y divides a number z, then x also divides z. In other words, if (x divides y) and (y divides z), then (x divides z).
In this case, we have (a divides b) and (b divides c). This means that (a divides c).
Therefore, for all integers a, b, and c, (a divides b) and (b divides c) implies a divides c.
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A ladder is leaned up against a wall. The bottom of the
ladder is 5 feet from the wall. The top of the ladder
reaches a spot on the wall that is 12 feet off the ground.
How long is the ladder? (HINT: draw a rough sketch!)
7 feet
13 feet
17 feet
20 feet
i
Answers
Answer:
Step-by-step explanation:
so 5 feet as a width and 12 as a height meaning that it would be b 13
A die is rolled. Find the probability of the given event. (a) The number showing is a 2; The probability is : (b) The number showing is an even number; The probability is : (c) The number showing is greater than 4; The probability is :
Answers
Answer:
(a) 1/6
(b) 1/2
(c) 1/3
Step-by-step explanation:
There are 6 possible outcomes.
(a) The number is 2: only 1 outcome out of 6, hence 1/6
(b) The number is even: {2, 4, 6} - 3 outcomes out of 6, hence 3/6 = 1/2
(c) The number is greater than four: {5, 6} - 2 outcomes out of 6, hence 2/6 = 1/3
Class A has 25 pupils and class B has 28 pupils.
Both classes sit the same maths test.
The mean score for class A is 69.
The mean score for both classes is 41.
What is the mean score in the maths test for class B?
Please helppp
Answers
Answer:
I think it will be less than 69 because the tozal mean is 41 while of A is 69 mean the values has decreased
Find the x-intercept of the line that is perpendicular to the
line 5x-25y+10=0 and passes through the point A(10,5).
Answers
Answer:
y=-5x+6
Step-by-step explanation:
The Slope is -5. And the Perpendicular slope is 1/5. Perpendicular equation y=mx+c Find c put in (0,-10) so -10=0/5+c so c=-10. so Perp eq is y=x/5–10 or x-5y=50
Show that w is in the subspace of R4 spanned by V1, V2, and V3, where these vectors are defined as follows. 1 -4 -9 -4 6 3 5 W= -2 - 1 - 1 -2 4 11 -8 - 15 To show that w is in the subspace, express was a linear combination of V1, V2, and V3. Select the correct answer below and, if necessary, fill in any answer boxes to complete your choice. O A. The vector w is in the subspace spanned by V1, V2, and V3. It is given by the formula w= (v1+ 2+ 3. (Simplify your answers. Type integers or fractions.) B. The vector w is not in the subspace spanned by V1, V2, and V3.
Answers
To show that w is in the subspace of R4 spanned by V1, V2, and V3, we need to find constants c1, c2, and c3 such that:
w = c1V1 + c2V2 + c3V3
We can write this as a matrix equation:
| 1 -4 -9 -4 | | c1 | | -2 |
| 6 3 5 1 | x | c2 | = | -1 |
| 2 4 11 -8 | | c3 | | -1 |
| -15 7 22 -14 | | -2 |
We can solve this system of equations using row reduction:
| 1 -4 -9 -4 | | c1 | | -2 |
| 6 3 5 1 | x | c2 | = | -1 |
| 2 4 11 -8 | | c3 | | -1 |
| -15 7 22 -14 | | -2 |
R2 = R2 - 6R1
R3 = R3 - 2R1
R4 = R4 + 15R1
| 1 -4 -9 -4 | | c1 | | -2 |
| 0 27 59 25 | x | c2 | = | 11 |
| 0 12 29 -16 | | c3 | | 3 |
| 0 -23 67 -59 | | -32 |
R4 = R4 + 23R2
| 1 -4 -9 -4 | | c1 | | -2 |
| 0 27 59 25 | x | c2 | = | 11 |
| 0 12 29 -16 | | c3 | | 3 |
| 0 0 174 -294 | | 225 |
R3 = R3 - (12/27)R2
R4 = (1/174)R4
| 1 -4 -9 -4 | | c1 | | -2 |
| 0 27 59 25 | x | c2 | = | 11 |
| 0 0 -1 22/3 | | c3 | | -13/3 |
| 0 0 1 -98/87 | | 25/58 |
R1 = R1 + 4R3
R2 = R2 - 59R3
R4 = R4 + (98/87)R3
| 1 0 -13 -10/3 | | c1 | | 21/29 |
| 0 27 0 -2119/87 | x | c2 | = | 2238/87 |
| 0 0 1 -98/87 | | c3 | | 25/58 |
| 0 0 0 1390/2391 | | 1009/2391 |
R1 = R1 + (13/1390)R4
R2 = (1/27)R2
R3 = R3 + (98/1390)R4
| 1
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What are the values of x and y in the figure
Answers
Answer:
I'm pretty sure 8
Step-by-step explanation:
1. If tan theta < 0 and sec theta > 0, which quadrant(s) could the terminal side of theta lie?
2. If csc theta > 0, which quadrant(s) could the terminal side of theta lie?
3. If sin theta < 0 and cot theta < 0, which quadrant(s) could the terminal side of theta lie?
I need help really quick, thank you to whoever can help! :)
Answers
If tan theta < 0 and sec theta > 0, the terminal side of theta could lie in either the second quadrant or the fourth quadrant.
If csc theta > 0, the terminal side of theta could lie in either the first quadrant or the second quadrant.
If sin theta < 0 and cot theta < 0, the terminal side of theta could lie in either the third quadrant or the fourth quadrant.
1. If tan theta < 0 and sec theta > 0, the terminal side of theta could lie in either the second quadrant or the fourth quadrant. This is because tan theta is negative in the second and fourth quadrants, and sec theta is positive in the first and fourth quadrants.
2. If csc theta > 0, the terminal side of theta could lie in either the first quadrant or the second quadrant. This is because csc theta is positive in the first and second quadrants.
3. If sin theta < 0 and cot theta < 0, the terminal side of theta could lie in either the third quadrant or the fourth quadrant. This is because sin theta is negative in the third and fourth quadrants, and cot theta is negative in the second and third quadrants.
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Chelsea is making a kite in the shape of a triangle. To determine if the triangle is a right triangle, Chelsea completed the following steps.
Step 1:
Find the side lengths of the triangle: 30 inches, 24 inches, 18 inches.
Step 2:
Substitute the values into the Pythagorean theorem: 18 squared + 24 squared = 30 squared.
Step 3:
Combine like terms: (18 + 24) squared = 30 squared.
Step 4:
Evaluate each side: 1764 not-equals 900.
Chelsea says the triangle is not a right triangle. Which best describes the accuracy of her explanation?
The triangle is actually a right triangle. In step 2, Chelsea incorrectly substituted the values into the Pythagorean theorem.
The triangle is not a right triangle, but in step 2 Chelsea incorrectly substituted the values into the Pythagorean theorem.
The triangle is actually a right triangle. In step 3, Chelsea incorrectly rewrote the expression on the left side of the equation.
The triangle is not a right triangle, but in step 3, Chelsea incorrectly rewrote the expression on the left side of the equation.
Answers
Answer:C
Step-by-step explanation:it’s really simple
Answer:
C.
Step-by-step explanation:
I am confused on this problem and am looking for help. If anyone could help me it would be appreciated
Answers
Answer
\(g(x)=-2(x+5)^{2}-3\)0. Horizontal translation 5 units.
,1. Reflection over the x-axis
,2. Vertical compression 2 units
,3. Vertical translation down 3 units
\(y=-(x+5)^2\)Explanation
• Writing the function in completed-square form.
As a ≠ 1, where a is the coefficient of the leading term, to write it in the completed-square form we have to make a = 1:
\(g(x)=-2x^2-20x-53\)\(g(x)=-2(x^2+10x+\frac{53}{2})\)Now we have to take half of the x term and square it and add it to the function as follows:
\(\frac{10}{2}^2=5^2=25\)\(g(x)=-2((x^2+10x+25)+\frac{53}{2}-25)\)Finally, we have a Perfect Squared Trinomial in the left side that we can rewrite as follows, obtaining our function g(x):
\(g(x)=-2(x+5)^2+\frac{3\cdot-2}{2}\)\(g(x)=-2(x+5)^2-3\)As our parent function is:
\(f(x)=x^2\)Then, the transformations that suffered were:
• Horizontal translation to the left 5 units
\(y=(x+5)^2\)• Reflection over the x-axis
\(y=-(x+5)^2\)• Vertical compression 2 units
\(y=-2(x+5)^2\)• Vertical translation down 3 units
\(g(x)=-2(x+5)^{2}-3\)PLEASE HELP!!! Identify the domain and range of the function graphed below.
Answers
The domain and the range of the function are listed below:
x ≥ - 4 All real numbers.How to find the domain and the range of the function
Herein we find a representation of the function, of which we need to determine graphically its domain and its range. The domain is the set of x-values, the values of the "horizontal" axis, and the range is the set of y-values, the values of the "vertical" axis. First, determine the domain of the function:
x ≥ - 4
Second, determine the range of the function:
All real numbers.
The domain and the range of the function are x ≥ - 4 and all real numbers, respectively.
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On a piece of paper graph
f(x) = {-1 if -1 <= x < 0}
{ -2 if 0 <= x < 0}
{ -3 if 1 <= x < 2
Answers
In this question, a piece-wise function is asked to be graphed.
Piece-wise function:
A piece-wise function is a function that has different definitions, depending on the input.
In this question, the function has three different definitions:
For x between -1(inclusive) and 0, y takes a constant value of -1.
For x between 0(inclusive) and 1, y takes a constant value of -2.
For x between 1(inclusive) and 2, y takes a constant value of -3.
Additionally:
At the inclusive points, the interval is circled, and thus, the graphic is given at the end of this answer.
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Answer:
Step-by-step explanation:
1. (1, 3); (2, 3)
2. (0, 0) (1, 0)
3. (-1, 1) (0,0)