The maximum value of [tex]\(f(x, y)\)[/tex] is given by:
[tex]\(f\left(\frac{b}{a+b}, \frac{a}{a+b}\right) = \left(\frac{b}{a+b}\right)^a \left(\frac{a}{a+b}\right)^b\)[/tex]
To find the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint (x + y = 1, use the method of Lagrange multipliers.
Let's set up the Lagrangian function:
[tex]\(L(x, y, \lambda) = x^a y^b + \lambda(x + y - 1)\)[/tex]
Taking the partial derivatives and setting them to zero:
Equation 1: [tex]\(\frac{\partial L}{\partial x} = a x^{a-1} y^b + \lambda = 0\)[/tex]
Equation 2: [tex]\(\frac{\partial L}{\partial y} = b x^a y^{b-1} + \lambda = 0\)[/tex]
Equation 3: [tex]\(\frac{\partial L}{\partial \lambda} = x + y - 1 = 0\)[/tex]
From equations (1) and (2) it can be written as
Equation 4: [tex]\(a x^{a-1} y^b = -\lambda\)[/tex] ... (4)
Equation 5: [tex]\(b x^a y^{b-1} = -\lambda\)[/tex] ... (5)
Dividing equation (4) by equation (5) gives
[tex]\(\frac{a}{b} \frac{x^{a-1}}{x^a} \frac{y^b}{y^{b-1}} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
From equation (3) [tex]\(y = 1 - x\)[/tex].
Substituting [tex]\(y = 1 - x\)[/tex] into the equation [tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{1-x}{x} = 1\)[/tex]
[tex]\(a x = b (1-x)\)[/tex]
[tex]\((a + b) x = b\)[/tex]
[tex]\(x = \frac{b}{a+b}\)[/tex]
Substituting [tex]\(x = \frac{b}{a+b}\)[/tex] into the constraint equation (3):
[tex]\(y = 1 - x[/tex]
[tex]= 1 - \frac{b}{a+b} = \frac{a}{a+b}\)[/tex]
Therefore, the critical point (x, y) that satisfies the constraint equation is:
[tex]\(x = \frac{b}{a+b}\) and \(y = \frac{a}{a+b}\)[/tex]
Therefore, the critical point gives the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint x + y = 1.
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how do you make a equation from slope intersecpt
coat that usually cost $123 is marked 1/3 off what is the sale price of the coat
Determine whether these statements are true or false.
a.∅ ∈ {∅}
b.∅ ∈ {∅,{∅}}
c.{∅} ∈ {∅}
d.{∅} ∈ {{∅}}
e.{∅} ⊂ {∅,{∅}} f ) {{∅}} ⊂ {∅,{∅}} g) {{∅}} ⊂ {{∅},{∅}}
a, b, d, e true: empty set fits in its own container.
c, f false: bigger container doesn't hold smaller container.
Here are the truths and falsehoods of the statements:
a) ∅ ∈ {∅} - True. The empty set is an element of the set containing only the empty set.
b) ∅ ∈ {∅, {∅}} - True. The empty set is also an element of the set containing both the empty set and the set containing the empty set.
c) {∅} ∈ {∅} - False. The set containing the empty set is not an element of the set containing only the empty set.
d) {∅} ∈ {{∅}} - True. The set containing the empty set is an element of the set containing only the set containing the empty set.
e) {∅} ⊂ {∅, {∅}} - True. The set containing the empty set is a subset of the set containing both the empty set and the set containing the empty set.
f) {{∅}} ⊂ {∅, {∅}} - False. The set containing the set containing the empty set is not a subset of the set containing only the empty set and the set containing the empty set.
The probable question can be: Determine whether these statements are true or false.
a) ∅ ∈ {∅}
b) ∅ ∈ {∅, {∅}}
c) {∅} ∈ {∅}
d) {∅} ∈ {{∅}}
e) {∅} ⊂ {∅, {∅}}
f) {{∅}} ⊂ {∅, {∅}}
I need help for all of these. Please give me all these answers, not just one please! :(
factor -1/4 out of -1/2-5/4y
To factor -1/4 out of -1/2-5/4y, you can use the distributive property and simplify the expression to 1/8 + 5/16y.
Explanation:To factor -1/4 out of -1/2-5/4y, you can use the distributive property. First, rewrite the expression as (-1/2) + (-5/4)y. Then, factor out -1/4: (-1/4)(-1/2) + (-1/4)(-5/4)y. This simplifies to 1/8 + 5/16y.
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Consider f and c below. f(x, y) = x2 i + y2 j c is the arc of the parabola y = 2x2 from (−1, 2) to (1, 2) (a) find a function f such that f = ∇f. f(x, y) = x3 3+ y3 3 correct: your answer is correct. (b) use part (a) to evaluate c ∇f · dr along the given curve
c.
In this exercise we have to perform the parameterization of the given function and we will have:
[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]
In there is some scalar function [tex]f(x, y)[/tex] such that:
[tex]\nabla f(x,y) = f(x, y) = x^2i+y^2j[/tex]
Then we want to find [tex]f[/tex] such that:
[tex]f'(x)= x^2= f(x,y) = \frac{x^3}{3} + g(y)\\f'(y)= y^2= f(x,y)= \frac{y^3}{3} + C\\f(x, y)= \frac{x^3}{3} +\frac{y^3}{3} + C[/tex]
So the vector filed [tex]f(x, y)[/tex] is conservative, which means the fundamental theorem applies; the line integral of [tex]f[/tex] along any path [tex]C[/tex] parameterized by some vector-valued function [tex]r(t)[/tex] over [tex]a\leq t\leq b[/tex] is given by:
[tex]\int\limits_C {f} \, dr\\\int\limits^t=a_t=b {f(r(t))} \, dt= f(r(b))-f(r(a))[/tex]
In the case:
[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]
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This problem requires calculation in multivariable calculus. Function f = ∇f when f(x, y) = x³/3 + y³/3. To calculate the line integral of ∇f · dr over the curve c, take the difference between the end and start points of the scalar function.
Explanation:The subject for this calculation is multivariable calculus, dealing with operations such as the gradient (∇) and line integrals.
According to the given function f(x, y) in part (a), it can be seen that f = ∇f for f(x, y) = x³/3 + y³/3. The vector field of this function is conservative since it corresponds to the gradient of a scalar function.
For part (b), you will need to calculate the line integral of ∇f · dr over the curve c, which is y = 2x² from (-1, 2) to (1, 2). The result of this will be the difference between the end and start points of the scalar function: f(1, 2) - f(-1, 2).
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Which of the following expressions could be used to represent "the difference between -17 and -8"?
-8 - (-17)
-17 - 8
-17 - (-8)
-8 + (-17)
The expression that represents "the difference between -17 and -8" is the second option:
-17 - (-8)
This expression translates to "negative seventeen minus negative eight," which yields the difference between the two numbers.
Expression: -8 - (-17):This expression represents "negative eight minus negative seventeen." When you subtract a negative number, it's equivalent to adding its positive counterpart. So, this expression can be simplified to:
-8 + 17 = 9
However, this result does not represent the difference between -17 and -8; it gives the difference between 17 and 8.
Expression: -17 - 8:This expression represents "negative seventeen minus eight." This directly calculates the difference between -17 and -8. It simplifies to:
-17 - 8 = -25
This indeed represents the difference between -17 and -8.
Expression: -17 - (-8):This expression represents "negative seventeen minus negative eight." Similar to the first expression, subtracting a negative number is the same as adding its positive counterpart. So, this simplifies to:
-17 + 8 = -9
This result does not represent the difference between -17 and -8; it gives the difference between -17 and 8.
Expression: -8 + (-17):This expression represents "negative eight plus negative seventeen." It simplifies to -25, which again represents the difference between -17 and -8.
Among these expressions, only the second expression, -17 - 8, correctly represents the difference between -17 and -8.
4 and one half divided by 3
ANSWER ASAP!!
A fair coin is tossed 9 times. what is the probability that the coin lands head at least 7 times?
a car drives 630 miles on 35 gallons of gas. How far can it drive on 12 gallons?
The car can drive 216 miles on 12 gallons.
What is displacement?The rate of change of position is called as displacement.
Now it is given that,
Distance traveled by car = 630 miles
Gas required to travel 630 miles = 35 gallons
Therefore, Distance travel n 1 gallon = Distance traveled by car / Gas required to travel 630 miles
⇒ Distance travel n 1 gallon = 630 / 35 miles
⇒ Distance travel n 1 gallon = 18 miles
Therefore, Distance travel n 12 gallon = 12 * Distance travel n 1 gallon
⇒ Distance travel n 12 gallon = 12 * 18 miles
⇒ Distance travel n 12 gallon = 216 miles.
Thus, the car can drive 216 miles on 12 gallons.
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Write an inequality for each situation fewer than 250 people atend the rally
What is the solution to the following system of equations? 3x-4y=35 and 3x+4y=5
The solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]
To solve the system of equations (3x - 4y = 35) and (3x + 4y = 5), we'll use the method of elimination.
Add the two equations together to eliminate the variable (y).
(3x - 4y) + (3x + 4y) = 35 + 5
3x - 4y + 3x + 4y = 40
6x = 40
[tex]\[ x = \frac{40}{6} \][/tex]
[tex]\[ x = \frac{20}{3} \][/tex]
Substitute [tex]\(x = \frac{20}{3}\)[/tex] into one of the original equations. Let's use the first equation:
[tex]\[ 3\left(\frac{20}{3}\right) - 4y = 35 \][/tex]
20 - 4y = 35
-4y = 35 - 20
-4y = 15
[tex]\[ y = \frac{15}{-4} \][/tex]
[tex]\[ y = -\frac{15}{4} \][/tex]
So, the solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]
The length of a rectangular garden is 7 feet longer than its width. The garden's perimeter is 202 feet. Find the width of the garden.
The formula B=32A is used in New England to estimate the minimum furnace output, B, in BTUs, for a modern house with A square feet of flooring. Determine the minimum furnace output for a 1700 square foot modern house
How is .8333 as a fraction 5/6?
The decimal 0.8333 is equivalent to the fraction 2/3.
To convert the decimal 0.8333 to a fraction, follow these steps:
Step 1: Let x be the decimal value:
x = 0.8333
Step 2: Multiply both sides of the equation by 10000 to remove the decimal:
10000x = 8333.3
Step 3: Subtract x from both sides of the equation:
10000x - x = 8333.3 - 0.8333
9999x = 8332.4667
Step 4: Divide both sides of the equation by 9999 to isolate x:
x = 8332.4667 / 9999
Divide both the numerator and the denominator by their greatest common divisor, which is 3333:
x = (8332.4667 ÷ 3333) / (9999 ÷ 3333)
x = (2.49995) / 3
Now, the fraction as a whole number over the denominator:
x = 2 / 3
Therefore, the fraction is 2/3.
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What is the domain of this relation?
{ (-1,5), (4,1), (8,3), (4,5) }
Gabriella bought three cantaloupe for seven dollars how many cantaloupes Shanya buy if she has $21
On a map of Texas, one inch represents 25 miles. If Dallas and San Antonio are 6.4 inches apart, how many miles apart are they? A) 1,600 B) 160 C) 256 D) 390.6
I believe your answer is B)160. But i'm not 100%. <3
Each member of a four-member relay team ran 100 meters during the relay. The time, in seconds, for each member is shown below. 13.9 s, 12.85 s, 14.82 s, 12.74 s Which expression finds the total time for the relay team?
Answer:
T = 13.9 + 12.85 + 14.82 + 12.74
Step-by-step explanation:
Great question, it is always good to ask away and get rid of any doubts that you may be having.
Based on the information in the question, the total time for the relay would be calculated by simply adding each of the four-relay members time during the 100 meter run. Therefore the expression would be
T = 13.9 + 12.85 + 14.82 + 12.74
T = 54.31 seconds.
If we decided to solve for the Total time (T) it would add up to 54.31 seconds for the relay time. But since the question did not ask for the total but just the expression. Then the answer is the expression above.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Using the Rational Root Theorem, what are all the rational roots of the polynomial f(x) = 20x4 + x3 + 8x2 + x – 12?
-4/5 and 3/4
-4/5 and -3/4
-1, -4/5, 3/4 and 1
-1. -4/5, -3/4 and 1
Answer:
Option 1 is correct.
Step-by-step explanation:
The given polynomial is
[tex]f(x)=20x^4+x^3+8x^2+x-12[/tex]
we have to find all the rational roots of the polynomial f(x)
The Rational Root Theorem states that the all possible roots of a polynomial are in the form of a rational number i.e in the form of [tex]\frac{p}{q}[/tex]
where p is a factor of constant term and q is the factor of coefficient of leading term.
In the given polynomial the constant is -12 and the leading coefficient is 20.
[tex]\text{All possible factor of -12 are }\pm1,\pm2, \pm3, \pm4,\pm6,\pm12[/tex]
[tex]\text{All possible factor of 20 are }\pm1,\pm2,\pm4,\pm5,\pm10,\pm20[/tex]
So, the all possible rational roots of the given polynomial are,
[tex]\pm1,\pm2, \pm3, \pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\pm\frac{1}{5},\pm\frac{3}{5},\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}[/tex]
Now, the rational roots of polynomial satisfy the given polynomial
[tex]f(-\frac{4}{5})=20(-\frac{4}{5})^4+(-\frac{4}{5})^3+8(-\frac{4}{5})^2-\frac{4}{5}-12=\frac{256}{625}\times 20-\frac{64}{125}+\frac{128}{125}-\frac{4}{5}-12[/tex]
[tex]=\frac{1024}{125}-\frac{64}{125}+\frac{128}{25}-\frac{4}{5}-12[/tex]
[tex]=\frac{960}{125}+\frac{128}{25}-\frac{4}{5}-12=12-12=0[/tex]
Hence, rational root.
[tex]f(\frac{3}{4})=20(\frac{3}{4})^4+(\frac{3}{4})^3+8(\frac{3}{4})^2+\frac{3}{4}-12=\frac{405}{64}+\frac{27}{64}+\frac{9}{2}+\frac{3}{4}-12=0[/tex]
rational root
[tex]f(1)=20(1)^4+(1)^3+8(1)^2+1-12=20+1+8-11=18\neq 0[/tex]
not a rational root.
hence, option 1 is correct
Answer:
The first option
Step-by-step explanation:
Right on edg:)
What is the term for a descriptive value found by using data from a census?
choices are....
Statistics
Parameter
Mean
Median
Mode
Answer:
Parameter
Step-by-step explanation:
Express 15/(1 - 3x)2 as a power series by differentiating the equation below. what is the radius of convergence?
If f(x) is a continuous function defined for all real numbers, f(–10) = –2, f(–8) = 5, and f(x) = 0 for one and only one value of x, then which of the following could be that x value?
–7
–9
0
2
The only value of x where f(x) = 0 can be determined by looking at the function's behavior near the x-axis and using the Intermediate Value Theorem. In this case, the potential x value is -9.
The one and only value of x where f(x) = 0 is the point of the function that intersects the x-axis. Since the function is continuous, it must cross the x-axis at this point due to the Intermediate Value Theorem.
To determine the possible value of x, we look at the given function values at x = -10 and x = -8 and identify where the function changes sign from negative to positive. The value of x where this sign change happens is the possible x value.
From the provided choices, the only value that would make this sign change possible is -9. Therefore, -9 could be the value of x where f(x) = 0.
What makes the statement true? 7^-2=
A box contains 12 1/2 square feet of tiles and costs $40.21 what is the approximate price per square foot
The approximate cost per square foot for the tiles is $3.22 when rounded to the nearest cent.
To calculate the price per square foot of tiles, divide the total cost of the tiles by the total number of square feet. The total cost given is $40.21, and the total area is 12 1/2 square feet.
First, convert the mixed number to an improper fraction. 12 1/2 is the same as 25/2 square feet. Now, to find the cost per square foot, perform the following division:
Price per square foot = Total cost / Total area in square feet
Price per square foot = $40.21 / (25/2 square feet)
Multiply by the reciprocal:
Price per square foot = $40.21 / 1 x 2 / 25
Price per square foot = $40.21 x 2 / 25
Price per square foot = $80.42 / 25
Price per square foot = $3.2168
Therefore, the approximate cost per square foot for the tiles is $3.22 when rounded to the nearest cent.
The approximate price per square foot is $3.22.
To find the price per square foot, divide the total cost by the total square footage. In this case, the cost is $40.21 and the square footage is 12 1/2 square feet.
1. Cost per square foot = Total cost / Total square footage
2. Cost per square foot = $40.21 / 12.5 sq ft ≈ $3.22/sq ft
1. Convert 12 1/2 square feet to a decimal: 12.5 sq ft.
2. Divide the total cost by the total square footage: $40.21 / 12.5 sq ft ≈ $3.22/sq ft.
To calculate the cost per square foot, you need to divide the total cost by the total square footage. In this case, the total cost is $40.21, and the total square footage is 12 1/2 square feet. First, convert 12 1/2 square feet to a decimal, which is 12.5 sq ft. Then, divide the total cost by the total square footage:
Cost per square foot = $40.21 / 12.5 sq ft ≈ $3.22/sq ft.
Therefore, the approximate price per square foot is $3.22.
complete question
A box contains 12 1/2 square feet of tiles and costs $40.21 what is the approximate price per square foot
What is the value of 4x to the 3rd degree + 4x when x=4
A transversal must intersect two or more _____ lines.
Answer:
paralell lines, i think.
Step-by-step explanation:
Last year, Tammy had $20,000 to invest. She invested some of it in an account that paid 7% simple interest per year, and she invested the rest in an account that paid 5% simple interest per year. After one year, she received a total of $1060 in interest. How much did she invest in each account?
Help super important
Based on the figure below, what is the value of x?
A. 5
B. 7
C. 11
D. 15
four students volunteer at the hospital. Casey volunteers 20.7 hours,Danielle, two and three fourths, Javier 18 nine tenths, And Forrest twenty and eighteen twenty fiths who volunteerd the greatsest number of hours?