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Section 8: Lagrange Multipliers

Instructions

  • First, you should watch the concepts videos below explaining the topics in the section. 
  • Second, you should attempt to solve the exercises and then watch the videos explaining the exercises. 
  • Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
  • This is the last section so you should return to the main several variable calculus page  or the main workshop page when you have finished the material below.

Concepts

  • Explanation of Lagrange's Theorem and Lagrange multipliers
  • Finding the absolute maximum or absolute minimum values of \(z=f(x,y)\) subject to the constraint \(g(x,y)=k\)
 



If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Conceptual V1

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Find the extreme values of \(f(x,y)=3x+y\) subject to the constraint \(x^2+y^2=10.\)

    Absolute Maximum: \(z=10\)
    Absolute Minimum: \(z=-10\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Lagrange Multipliers Exercise V1


  2. Find the extreme values of \(f(,y)=x^2+2y^2\) subject to the constraint \(x^2+16y^2=16.\)

    Absolute Maximum: \(z=16\) 
    Absolute Minimum: \(z=2\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Lagrange Multipliers Exercise V2


  3. Find the minimum values of \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint \(x+3y-2z=12.\)

    Absolute Minimum: \(f\left(\frac{6}{7},\frac{18}{7},-\frac{12}{7}\right)=\dfrac{72}{7}\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Lagrange Multipliers Exercise V3


  4. Find the volume of the largest rectangular box with faces parallel to the coordinate planes that can be inscribed in the ellipsoid \(16x^2+4y^2+9z^2=144.\)

    The maximum volume is \(V=64\sqrt{3}\) units\(^3.\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Lagrange Multipliers Exercise V4


 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. How do we know when it is proper to use the method of Lagrange to find the absolute extrema of \(z=f(x,y)\)?
  2. What system of equations must be solved in order to find the absolute extrema of \(z=f(x,y)\) subject to \(g(x,y)=k\)?
  3. Choose an exercise from Section 7 that uses The Extreme Value Theorem, and instead use the method of Lagrange. Determine whether the method of Lagrange is superior over The Extreme Value Theorem method for the problem you chose.