Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Lagrange Multipliers

Study concepts, example questions & explanations for Calculus 3

Create An Account Create Tests & Flashcards

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 Next →

Example Question #7 :Applications Of Partial Derivatives

Find the minimum and maximum of f(x,y)=2x-5y , subject to the constraint x^2+y^2=144 .

Possible Answers:

f(4.46,11.14) is a maximum

f(-4.46,-11.14) is a minimum

f(-4.46,11.14) is a maximum

f(4.46,-11.14) is a minimum

There are no maximums or minimums

f(4.46,-11.14) is a maximum

f(-4.46,11.14) is a minimum

f(4.46,11.14) is a maximum

f(4.46,-11.14) is a minimum

Correct answer:

f(4.46,-11.14) is a maximum

f(-4.46,11.14) is a minimum

Explanation:

First we need to set up our system of equations.

$2=2\lambda x \rightarrow x=\frac{1}{\lambda}$

$-5=2\lambda y \rightarrow y=-\frac{5}{2 \lambda}$

x^2+y^2=144

Now lets plug in these constraints.

$(\frac{1}{\lambda})^2+(-\frac{5}{2\lambda})^2=144$

$\frac{1}{\lambda ^2}+\frac{25}{4\lambda^2}=144$

$\frac{29}{4\lambda^2}=144$

Now we solve for $\lambda$

$\frac{29}{4\lambda^2}=144\rightarrow \lambda^2=\frac{29}{576}\rightarrow\lambda=\pm\sqrt{\frac{29}{576}}$

如果

$\lambda=\sqrt{\frac{29}{576}}$

$x=\frac{1}{\sqrt{\frac{29}{576}}}\approx 4.46$ , $y=-\frac{5}{2\sqrt{\frac{29}{576}}}\approx -11.14$

如果

$\lambda=-\sqrt{\frac{29}{576}}$

$x=\frac{1}{\sqrt{\frac{29}{576}}}\approx -4.46$ , $y=-\frac{5}{2\sqrt{\frac{29}{576}}}\approx 11.14$

Now lets plug in these values of, andinto the original equation.

f(4.46,-11.14)=2(4.46)-5(-11.14)=64.62

f(-4.46,11.14)=2(-4.46)-5(11.14)=-64.62

We can conclude from this that f(4.46,-11.14) is a maximum, and f(-4.46,11.14) is a minimum.

Report an Error

Example Question #8 :Applications Of Partial Derivatives

Find the absolute minimum value of the function f(x,y) = x^2+y^2 subject to the constraint x^2 +2y^2 = 1 .

Possible Answers:

$\sqrt2$

$2\sqrt2$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

Let g = x^2 +2y^2 To find the absolute minimum value, we must solve the system of equations given by

$\triangledown f = \lambda\triangledown g, g =1$ .

So this system of equations is

$f_x = \lambda g_x$ , $f_y = \lambda g_y$ , g =1 .

Taking partial derivatives and substituting as indicated, this becomes

$2x = \lambda(2x), 2y = \lambda(4y), x^2+2y^2 = 1$ .

From the left equation, we see either x=0 or $\lambda = 1$ . If x=0 , then substituting this into the other equations, we can solve for $y, \lambda$ , and get $y = \pm \sqrt{2}/2$ , $\lambda = 1/2$ , giving two extreme candidate points at $(0, \frac{\sqrt{2}}{2}), (0, -\frac{\sqrt{2}}{2})$ .

On the other hand, if instead $\lambda =1$ , this forces y = 0 from the 2nd equation, and $x = \pm 1$ from the 3rd equation. This gives us two more extreme candidate points; (-1,0),(1,0) .

Taking all four of our found points, and plugging them back into, we have

$f(-1,0) = 1, f(1,0) = 1, f(0,\frac{\sqrt{2}}{2}) = \frac{1}{2}, f(0,-\frac{\sqrt{2}}{2}) = \frac{1}{2}$ .

Hence the absolute minimum value is $\frac{1}{2}$ .

Report an Error

Example Question #9 :Applications Of Partial Derivatives

Find the dimensions of a box with maximum volume such that the sum of its edges iscm.

Possible Answers:

$6 \times 5 \times 4$

$6 \times 6 \times 6$

$5 \times 4 \times 3$

$5 \times 5 \times 5$

Correct answer:

$5 \times 5 \times 5$

Explanation:

Untitled

Report an Error

Example Question #11 :Applications Of Partial Derivatives

Optimize $f(x)=\sqrt{6-x^2-y^2}$ using the constraint x+y-2=0

Possible Answers:

$\pm\frac{1}{2}$

$\pm2$

Correct answer:

$\pm2$

Explanation:

Untitled

Report an Error

Example Question #12 :Applications Of Partial Derivatives

Maximize f(x,y,z)=xyz with constraint x^2+2y^2+3z^2=6

Possible Answers:

$3\sqrt{2}$

$\frac{\sqrt{3}}{2}$

$\frac{3}{2\sqrt{2}}$

$\frac{3}{2}$

Correct answer:

$\frac{3}{2\sqrt{2}}$

Explanation:

Untitled

Report an Error

Example Question #13 :Applications Of Partial Derivatives

A company has the production function $P=100L^{\frac{3}{4}}K^{\frac{1}{4}}$ , whererepresents the number of hours of labor, andrepresents the capital. Each labor hour costs $150 and each unit capital costs $250. If the total cost of labor and capital is is $50,000, then find the maximum production.

Possible Answers:

$\$12000$

$\$12500$

none of the above

$\$12300$

Correct answer:

$\$12300$

Explanation:

Untitled

Report an Error

Example Question #14 :Applications Of Partial Derivatives

Find the maximum value of the function f=x^2y with the constraint x^2+y^2=4 .

Possible Answers:

$x=\frac{2\sqrt{6}}{3}$ , $y=\frac{2\sqrt{3}}{3}$

$x=\sqrt{3}$ , y=1

x=2 , y=2

x=1 , $y=\sqrt{3}$

Correct answer:

$x=\frac{2\sqrt{6}}{3}$ , $y=\frac{2\sqrt{3}}{3}$

Explanation:

To optimize a functionsubject to the constraint g=k , we use the Lagrangian function, $\bigtriangledown f=\lambda \bigtriangledown g$ , where $\lambda$ is the Lagrangian multiplier.

如果is a two-dimensional function, the Lagrangian function expands to two equations,

$f_x=\lambda g_x$ and $f_y=\lambda g_y$ .

The equation being optimized is f=x^2y .

The constraint is x^2+y^2=4 .

f_x=2xy , f_y=x^2 , g_x=2x , g_y=2y

Substituting these variables into the the Lagrangian function and the constraint equation gives us the following equations

$2xy=\lambda (2x)$

$x^2=\lambda (2y)$

x^2+y^2=4

We have three equations and three variables (,, and $\lambda$ ), so we can solve the system of equations.

$2xy=\lambda (2x)\Rightarrow y=\lambda$

$x^2=\lambda (2y)\Rightarrow x^2/2y=\lambda$

Setting the two expressions for $\lambda$ equal to each other gives us

y=x^2/2y

2y^2=x^2

Substituting this expression into the constraint gives us

x^2+y^2=4

2y^2+y^2=4

$3y^2=4\Rightarrow y^2=4/3$

$y=2/\sqrt{3}=\frac{2\sqrt{3}}{3}$

x^2=2y^2=2(4/3)=8/3

$x=\sqrt{\frac{8}{3}}=\frac{2\sqrt{2}}{\sqrt{3}}=\frac{2\sqrt{6}}{3}$

Report an Error

Example Question #15 :Applications Of Partial Derivatives

Find the maximum value of the function f=x^2+y^2-xy with the constraint 3x+4y=12 .

Possible Answers:

$x=\frac{60}{37}$ , $y=\frac{66}{37}$

x=2 , y=1.5

$x=\frac{110}{37}$ , $y=\frac{121}{37}$

$x=\frac{8}{3}$ , y=1

Correct answer:

$x=\frac{60}{37}$ , $y=\frac{66}{37}$

Explanation:

To optimize a functionsubject to the constraint g=k , we use the Lagrangian function, $\bigtriangledown f=\lambda \bigtriangledown g$ , where $\lambda$ is the Lagrangian multiplier.

如果is a two-dimensional function, the Lagrangian function expands to two equations,

$f_x=\lambda g_x$ and $f_y=\lambda g_y$ .

The equation being optimized is f=x^2+y^2-xy .

The constraint is 3x+4y=12 .

f_x=2x-y , f_y=2y-x , g_x=3 , g_y=4

Substituting these variables into the the Lagrangian function and the constraint equation gives us the following equations

$2x-y=\lambda (3)$

$2y-x=\lambda (4)$

3x+4y=12

We have three equations and three variables (,, and $\lambda$ ), so we can solve the system of equations.

$2x-y=\lambda (3)\Rightarrow 2x/3-y/3=\lambda$

$2y-x=\lambda (4) \Rightarrow y/2-x/4=\lambda$

Setting the two expressions for $\lambda$ equal to each other gives us

2x/3-y/3=y/2-x/4

2x/3+x/4=y/2+y/3

$\frac{11x}{12}=\frac{5y}{6}$

$x=\frac{10y}{11}$

Substituting this expression into the constraint gives us

3x+4y=12

3(10y/11)+4y=12

30y/11+4y=12

74y/11=12

$y=12*11/74=\frac{66}{37}$

$x=\frac{10}{11}*\frac{66}{37}=\frac{60}{37}$

Report an Error

Example Question #16 :Applications Of Partial Derivatives

A company makes chairs () and benches (). The profit equation for this company is f=-2c^2-3b^2+24c+28b+11 . The company can only producepieces per day. How many of each seat should the company produce to maximize profit?

Possible Answers:

b= 25 , c= 9

b= 17 , c= 18

b= 14 , c= 20

b= 8 , c= 27

Correct answer:

b= 14 , c= 20

Explanation:

To optimize a functionsubject to the constraint g=k , we use the Lagrangian function, $\bigtriangledown f=\lambda \bigtriangledown g$ , where $\lambda$ is the Lagrangian multiplier.

如果is a two-dimensional function, the Lagrangian function expands to two equations,

$f_x=\lambda g_x$ and $f_y=\lambda g_y$ .

In this problem, we are trying to maximize the profit, so the equation being optimized is f=-2c^2-3b^2+24c+28b+11 .

The company can only producepieces of furniture, so the constraint is c+b=35 .

f_c=-4c+24

f_b=-6b+28

g_c=1 , g_b=1

Substituting these variables into the the Lagrangian function and the constraint equation gives us the following equations

$-4c+24=\lambda (1)$

$-6b+28=\lambda (1)$

c+b=35

We have three equations and three variables (,, and $\lambda$ ), so we can solve the system of equations.

Setting the two expressions of $\lambda$ equal to each other gives us

-4c+24=-6b+28

6b-28=4c-24

6b-4=4c

c=6b/4-1

Substituting this expression into the constraint gives

c+b=35

6b/4-1+b=35

10b/4=36

$b=14.4 \approx 14$

$c=6(14.4)/4-1=20.6 \approx 20$

Profit is maximized by makingchairs andbenches.

Report an Error

Example Question #17 :Applications Of Partial Derivatives

A company makes end tables () and side tables (). The profit equation for this company is f=-7e^2-2s^2+10e+15s+55 . The company can only producepieces per day. How many of each table should the company produce to maximize profit?

Possible Answers:

e=15 , s=24

e=4 , s=35

e=25 , s=14

e=10 , s=39

Correct answer:

e=10 , s=39

Explanation:

To optimize a functionsubject to the constraint g=k , we use the Lagrangian function, $\bigtriangledown f=\lambda \bigtriangledown g$ , where $\lambda$ is the Lagrangian multiplier.

如果is a two-dimensional function, the Lagrangian function expands to two equations,

$f_x=\lambda g_x$ and $f_y=\lambda g_y$ .

In this problem, we are trying to maximize the profit, so the equation being optimized is f=-7e^2-2s^2+10e+15s+55 .

The company can only producepieces of furniture, so the constraint is e+s=50 .

f_e=-14e+10

f_s=-4s+15

g_e=1 , g_s=1

Substituting these variables into the the Lagrangian function and the constraint equation gives us the following equations

$-14e+10=\lambda(1)$

$-4s+15=\lambda (1)$

e+s=50

We have three equations and three variables (,, and $\lambda$ ), so we can solve the system of equations.

Setting the two expressions of $\lambda$ equal to each other gives us

-14e+10=-4s+15

4s-15=14e-10

4s=14e+5

s=7e/2+5/4

Substituting this expression into the constraint gives

e+s=50

e+7e/2+5/4=50

9e/2+5/4=50

9e/2=195/4

$e=65/6 \approx 10$

$s=\frac{7}{2}*\frac{65}{6}+\frac{5}{4}=\frac{455}{12}+\frac{5}{4}=\frac{470}{12} \approx 39$

Profit is maximized by makingend tables and副表。

Report an Error

← Previous 1 2 Next →

View Calculus 3 Tutors

Francisco
Certified Tutor

University of Texas at Brownsville, Bachelors, Mathematics. University of Texas at Brownsville, Masters, Mathematics.

View Calculus 3 Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Calculus 3 Tutors

Mehrdad
Certified Tutor

Tehran University, Mechanical Engineer, Mechanical Engineering. Amirkabir University, Doctor of Philosophy, Mechanical Engine...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ACT Tutors in Dallas Fort Worth,SSAT Tutors in Dallas Fort Worth,LSAT Tutors in Dallas Fort Worth,Math Tutors in Houston,SAT Tutors in San Diego,GMAT Tutors in Dallas Fort Worth,ISEE Tutors in Atlanta,SAT Tutors in Miami,Algebra Tutors in Los Angeles,ISEE Tutors in Miami

Popular Courses & Classes

Spanish Courses & Classes in Phoenix,GMAT Courses & Classes in Dallas Fort Worth,SSAT Courses & Classes in Seattle,Spanish Courses & Classes in Chicago,Spanish Courses & Classes in Washington DC,GMAT Courses & Classes in San Francisco-Bay Area,Spanish Courses & Classes in New York City,GRE Courses & Classes in Washington DC,ISEE Courses & Classes in Atlanta,SAT Courses & Classes in San Diego

Popular Test Prep

LSAT Test Prep in San Francisco-Bay Area,LSAT Test Prep in Chicago,SSAT Test Prep in Atlanta,GMAT Test Prep in Denver,SAT Test Prep in Dallas Fort Worth,SAT Test Prep in Denver,SAT Test Prep in Seattle,GMAT Test Prep in San Francisco-Bay Area,SSAT Test Prep in Seattle,SSAT Test Prep in Miami

DMCA Complaint

如果you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

*Full legal name

公司名称

*Phone number

*Email address

Address

*Address1

Address2

*City

*State

*Zipcode

*Country

Complaint Details

*URL of copyrighted work (where your original material is located)

*Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

*Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 3 : Lagrange Multipliers

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #7 :Applications Of Partial Derivatives

Example Question #8 :Applications Of Partial Derivatives

Example Question #9 :Applications Of Partial Derivatives

Example Question #11 :Applications Of Partial Derivatives

Example Question #12 :Applications Of Partial Derivatives

Example Question #13 :Applications Of Partial Derivatives

Example Question #14 :Applications Of Partial Derivatives

Example Question #15 :Applications Of Partial Derivatives

Example Question #16 :Applications Of Partial Derivatives

Example Question #17 :Applications Of Partial Derivatives

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:

Email address:
Your name:
Feedback: