Find the minimum and maximum values of the function subject to the given constraint. (if an answer does not exist, enter dne.) f(x, y) = x2y + x + y, xy = 5

Answers

Answer 1
Via Lagrange multipliers:

[tex]L(x,y,\lambd)=x^2y+x+y+\lambda(xy-5)[/tex]
[tex]L_x=2xy+1+\lambda y=0[/tex]
[tex]L_y=x^2+1+\lambda x=0[/tex]
[tex]L_\lambda=xy-5=0[/tex]

[tex]\underbrace{10}_{2xy}+1+\lambda y=0\implies \lambda=-\dfrac{11}y[/tex]
[tex]xy=5\implies y=\dfrac5x\implies\lambda=-\dfrac{11}5x[/tex]

[tex]\impliesx^2+1+\left(-\dfrac{11}5x\right)x=0\implies x^2=\dfrac56\implies x=\pm\sqrt{\dfrac56}[/tex]
[tex]xy=5\implies y=\pm\sqrt{30}[/tex]

At these points, we get local minima of [tex]f\left(\pm\sqrt{\dfrac56},\pm\sqrt{30}\right)=\pm2\sqrt{30}[/tex].

- - -

Another way to do this is to make [tex]f(x,y)[/tex] a function independent of [tex]y[/tex], which is made possible by the constraint.

[tex]xy=5\implies y=\dfrac5x[/tex]
[tex]\implies f(x,y)=f\left(x,\dfrac5x\right)=F(x)=6x+\dfrac5x[/tex]
[tex]\implies F'(x)=6-\dfrac5{x^2}=0\implies x=\pm\sqrt{\dfrac56}[/tex]

and so on.

Related Questions

Order these numbers from least to greatest. 14325 , 5.753 , 534 , 5.34

Answers

5.34, 5.753, 534, 14325

Deanna has 80 books in her collection. If 60% of the books are fiction, how many of the books are nonfiction?

Answers

if 60% is fiction, then 40% are non fiction

80 x 40% =

80 x 0.40 = 32

 32 books are nonfiction

Twenty different books are to be put on five book shelves, each of which holds at least twenty books.

Answers

Hello There!

The total number of different arrangements is 10,628.

Hope This Helps You!
Good Luck :) 

- Hannah ❤

Which of the following is a sample of the population of bookstores on the West Coast of the United States? I. All book stores in California II. Randomly selected children's book stores III. Internet book stores A. I only B. II only C. I and III D. III only E. I and II

Answers

Eliminate (I) "all bookstores in California," because we're choosing a SAMPLE, not an entire population of stores.

Keeping this in mind, we'd not choose "Internet bookstores" unless it's clear that we're sampling these stores, not looking at all of them.

This leaves only "randomly selected children's bookstores."  Even now I'm not comfortable, because this choice focuses specifically on children's stores (instead of on bookstores in general).

Answer:

It would be I - All bookstores in California

Step-by-step explanation:

Internet book stores and random children's bookstores do not specify that they're on the West Coast and could be taken from anywhere

Determine whether the fractions -4/5, -4/6, -4/7, and -4/8 are arranged in order from least to greatest. explain

Answers

They are ordered least to greatest

Translate the phrase to mathematical language. Then simplify the expression.

Subtract 81 from −55.
What is a numerical expression for the​ phrase?__
​(Do not​ simplify.)

Simplify the expression.__

Answers

"Subtract 81 from −55" can be written symbolically as (-55) - 81.

Simplifying, -55-81 = - 136

Is it possible to construct a triangle with side lengths of 11 cm, 15 cm, and 21 cm? Why or why not?

Answers

Assume the triangle is ABC with: AB = 11 ; AC = 15 and BC = 21.
To get a triangle each side should smaller than the addition of the other sides:

AB< AC+BC (11< 15+21) For this side it's ok
AC< AB+BC (15< 11+21) For this side it's ok
BC< AB+AC ( 21<11+15. This is wrong since 26=26 thus you can't draw a triangle with such measurements


what is the decimal of 77/200?

Answers

0.385 is the decimal
0.385 is the decimal. Use a calculator and divide the numerator by the denominator

When hired at a new job selling jewelry, you are given two pay options:

Option A: Base salary of $16,000 a year, with a commission of 7% of your sales

Option B: Base salary of $25,000 a year, with a commission of 2% of your sales


In order for option A to produce a larger income, you would need sell at least $_______ of jewelry each year

Answers

16,000 + 0.07x > 25,000 + 0.02x
0.07x - 0.02x > 25,000 - 16,000
0.05x > 9000
x > 9000 / 0.05
x > 180,000....so they would need to sell at least 180,001

You are required to sell a minimum of 180,001 of jewelry each year.

Given that,

The base salary is $16,000 per year and there is a 7% commission on sales for option A. The base salary is $25,000 per year and there is a 2% commission on sales for option B.Also, let us assume the sales be x

Based on the above information, the following expression should be made:

16,000 + 0.07x > 25,000 + 0.02x

0.07x - 0.02x > 25,000 - 16,000

0.05x > 9000

x > 9000 ÷ 0.05

x > 180,000

Therefore we can conclude that you are required to sell a minimum of 180,001 of jewelry each year.

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mr. renato had $150 in his bank account. he deposited $50. acuple of days later he withdrew $20. a week later he deposited $75 .the next week he withdrew $100. what is mr. Renato balance after all the transactions

Answers

He starts out with 150. 
150+ 50+200
200-20=180
180+75= 255
255-100= 155
After all the transactions, he had $155.

scuba diver is swimming 30 feet below sea level . A hand glider passes over the position of the scuba diver. The hand glider is 215 ft above sea level.
draw a number line to represent the distance between the scuba diver and the hand glider (include sea level at zero feet)

Answers

Attached the solution and work.

Without graphing, tell which statements are true for the graph of the function y = x2 + 2.

I. All points on the graph are above the origin.

II. All points on the graph have positive x-values.

III. All points on the graph have positive y-values.

Statements (... and ...) are true. Statement (...) is true because x^2 is non-negative for all values of x, so the minimum value of x^2 is 0. Thus, the minimum y−value for the function is (...) and the lowest point on the graph is (0, ...), which is above the origin. Statement (...) is true because the lowest point on the graph is (0, ...) so all other points have a y−value that is greater than (...) and must be positive.

Answers

The answers are choices one and three. Hope this helps!
Final answer:

Statements I and III are true as per the defined function y=x^2+2. The graph of this function would have all points above the origin and all points would have positive y-values. Statement II is incorrect as x can take both negative and positive values.

Explanation:

Considering the three statements given, only the statements I and III are true. This is because for the function y = x^2 + 2, all points on the graph are above the origin and all points on the graph have positive y-values.

Statement I is true because the function x^2 is non-negative for all values of x, which means that the minimum value of x^2 is 0. As a result, the minimum y−value for the function is 2 when x=0 and the lowest point on the graph is (0, 2), which is indeed above the origin.

Statement III is true because the lowest point on the graph is (0, 2) and since 2 is positive all other points have a y−value that is greater than 2 and therefore must be positive. Statement II is not true as x can take both negative and positive values in the function y=x^2+2.

Learn more about quadratic function here:

https://brainly.com/question/35505962

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The volume of this rectangular prism is 6x3. What does the coefficient 6 mean in terms of the problem?

A. The total volume of the prism is 6 cubic units.
B. The length is 6 times the width.
C. The width is 6 times the height and the length.
D. The length of the prism is 6 units.

Answers

Answer:

B. The length is 6 times the width

Step-by-step explanation:

The proper solution is choice B. The length is 6 times the width.

How do you resolve rectangular prisms?

To find the quantity of a rectangular prism, multiply its 3 dimensions: duration x width x height. The volume is expressed in cubic gadgets.

What are examples of square prisms?

Right rectangular prisms or cuboids are all around us. a few examples are books, packing containers, homes, bricks, forums, doorways, containers, cabinets, mobiles, and laptops.

What does the coefficient mean?

Any of the elements of a product taken into consideration with regards to a specific element particularly: a constant thing of a term as outstanding from a variable.

Learn more about rectangular prisms here: https://brainly.com/question/477459

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A soccer coach purchased one goal and some soccer balls for the team. The expression 180 + 15x represents the cost before tax. What do the different parts of the expression model? Drag the parts of the expression into the boxes to match each description.

Answers

The expression [tex]180+15x[/tex] consists of ONE CONSTANT [180] and another VARIABLE EXPRESSION [15x].

Since the coach purchased ONE GOAL, this corresponds to the CONSTANT, which is 180. 180 is the price of the goal.

Since coach purchased several soccer balls, but doesn't say how many, it is the variable part, 15x, where

either 15 is price of ball and x is the number of balls, OR15 is the number of balls and x is the price of each

ANSWER:

180 represents the price of the goal

15x represents the total price of the soccer balls

Answer:

180 represents the price of the goal

15x represents the total price of the soccer balls

Step-by-step explanation:

FREE POINTS!!!!!!!!!! 10 POINTS Len's recipe for blueberry muffins calls for 110.25 grams of brown sugar for the muffin batter and an additional 55.45 grams of brown sugar for the topping. Len wants to make 4.5 batches of muffins. Enter the total amount of brown sugar that Len will need

Answers

Wow.... My brother Said that it took him a while to figure it out... but

add 110.25 and 55.45

you would get 165.70

multiply 165.70 by 4.5

then you would get 745.65

745.65 is your answer 

the answer is 745.65

hope i helped:)

ABCD is a trapezoid. = 2.5 cm, and = 4 cm, and the area = 6.5 cm2. What is the altitude of the trapezoid?

Answers

6.5 = 1/2(2.5 + 4) h
6.5 = 1/2(6.5)h
6.5 = 3.25 h
h = 6.5 / 3.25
h = 2

answer
the altitude of the trapezoid is 2 cm

Answer:

Altitude of the trapezoid is 2 cm.

Step-by-step explanation:

ABCD is a trapezoid. Since two bases of the trapezoid are

B1 = 2.5 cm and B2 = 4 cm.

Area given is 6.5 cm²

Then we have to calculated he altitude of the trapezoid.

Now we know the formula of area of the trapezoid = [tex]\frac{1}{2}[/tex] ( Base1 + Base2)

By putting the values in the formula

6.5 =  [tex]\frac{1}{2}[/tex] ( 2.5 + 4 ) × h

6.5 =  [tex]\frac{6.5}{2}[/tex] × h

h =  \frac{2\times 6.5}{6.5} = 2 cm.

Answer is altitude of the trapezoid is 2 cm.

What is the interquartile range of 4 5 7 9 10 14 16 24

Answers

2's and 3's of GCF


HOpe iT hElpeED I'M cOOl

Answer:

The answer is 9

Step-by-step explanation: ? i did the the tetst or whatever and 9 is the correct answer.

YOUR WELCOME :}

katie and carol bought toys for charity toy drive.katie bought 125 dolls for 13$ each. carol bought 65 toy boats for 32$ each.who spent more money altogher

Answers

the answer is carol
125 x 13 = 1625
65 x 32 = 2080
take 125* 13= 1625 Katie spending
then
take 65*32= 2080 Carol spending

select the graph that represents the given set a={(1,2),(2,3),(3,4),(4,5)

Answers

y=x+1 is the equation for the graph

the GCF of the following numbers. 32 and 40

Answers

The answer should be 8.

Shelly used exactly 1/ 3 of a cup of oil and 3/ 8 of a cup of water in a recipe. How many cups of liquid did Shelly use in all?

Answers

the answer would be 17/24 cups of liquid. hope that helped

Answer: =[tex]\dfrac{17}{24}[/tex]

Step-by-step explanation:

Given : The amount of oil is used  in a recipe. = [tex]\dfrac{1}{3}[/tex]

The amount of water used in a recipe = [tex]\dfrac{3}{8}[/tex]

The total number of cups of liquid used by Shelly:-

[tex]\dfrac{1}{3}+\dfrac{3}{8}=\dfrac{8+3\times3}{24}\\\\\=\dfrac{8+9}{24}=\dfrac{17}{24}[/tex]

Hence, the total number of cups of liquid used in all =[tex]\dfrac{17}{24}[/tex]

Carlos has 7 times as many red toy cars as blue toy cars. He has 56 total cars. How many more red cars does he have?

Answers

let x = blue cars

 so 7x = red cars

x +7x = 56

8x = 56

x = 56/8 =7 blue cars

7*7 =49 red cars

49-7 =42 more red cars

Th local park has 4 bike racks. Each bike rack can hold 15 bikes. There are 16 bikes in the bike racks. What expression shows the total number of empty spaces in the bike racks?

Answers

4 * 15 bike storage capacity

16 spaces are occupied

This leaves 60 - 16, or 44, spaces available.

what is -8 3/4 + -11 1/3

Answers

The answer to this is -29/3, or -9.6...

Hope this helped!! ☺♥
The answer is -20 1/12.

Hope this helps! :D

~PutarPotato

Jason bought 5 movie tickets and a bag of popcorn Jeremy bought 2 movie tickets they spent 54$ total. If the popcorn cost 5$, how much did each movie ticket cost?

Answers

total tickets: 7
Popcorn costs: $5
total cost: $54

7x + 5 = 54
     - 5     -5

7x = 49

49/7 = 7

Final Answer: x=7

Differentiate f and find the domain of f. (enter the domain in interval notation.)f(x) = ln(x2 − 12x)derivative f '(x) = 2x−12x2−12x​ correct: your answer is correct.domain

Answers

f(x) = ln(x² - 12x)
The derivative is
[tex]f'(x)= \frac{2x-12}{x^{2}-12x} [/tex]

f(x) is undefined when x² - 12x = x(x - 12) = 0
That is, when x = 0 or x = 12.
Therefore the domain is (-∞, 0)∪(0,12)∪(12, ∞)

Answer:
The derivative is
[tex]f'(x) = \frac{2x-12}{x^{2}-12x} [/tex]
The domain is
(-∞, 0)∪(0, 12)∪(12,∞)



write the number 1,744,326 in expanded form

Answers

1,000,000 + 700,000 + 40,000 + 4,000 + 300 + 20 + 6
one million, seven hundred forty-four thousand, three hundred twenty-six.

The median for the given set of 6 ordered data values is 32.5.
9, 12, 25, _, 41, 48
What is the missing value?

Answers

Being that u have a set of data that has an even number of numbers, the median (middle number) would be found by adding the 2 middle numbers and dividing by 2.

ur 2 middle numbers would be 25 and x...and when added, u divide by 2, u will get 32.5...so lets set it up as an equation.

(25 + x) / 2 = 32.5...multiply both sides by 2
25 + x = 32.5 * 2
25 + x = 65
x = 65 - 25
x = 40 <== ur missing number

how to solve an equation for x with natural log of x

Answers

Question is not specific, so can only give a general answer.
As an example:
If the equation is of the form
log(x)=5
then raise each side to the power of e to give
e^(log(x))=e^5
Since e^(log(x))=x, we have
x=e^5.

A logarithmic equation involving natural logarithms can be solve by taking the exponent of both sides

How to solve a natural logarithmic equation for x

To solve an equation involving the natural logarithm of x, you generally follow these steps:

Isolate the natural logarithm termExponentiate both sidesSolve for x: After removing the natural logarithm, solve the resulting equation for x.

Take for instance, the equation is of the form

2 + ln(x) = 7

Isolate the natural logarithm

ln(x) = 5

Take the exponent of both sides

So, we have

[tex]x = e^5[/tex]

Evaluate the exponent

x = 148.41

Hence, the logarithmic equation solved for x is x = 148.41

Which BEST describes the construction of a triangle if given one side length of 10 cm and one right angle?
A)
Unique triangle


B)
More than one triangle


C)
Triangle not possible


D)
Cannot be determined

Answers

B, is the answer! :))))))))))))))

Answer:

b

Step-by-step explanation:

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