Scroll back to the top

Virtual Math Learning Center Logo

Section 7: Maximum and Minimum Values

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.
  • When you have finished the material below, you can start onthe next section or return to the main several variable calculus page.

Concepts

  • Local and absolute extrema of a function \(z=f(x,y)\)
  • The Second Derivative Test for Local Extrema
  • Extreme Value Theorem for Functions of Two Variables
 



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.
Maximum and Minimum Value (multivariable) Conceptual V1


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.
Maximum and Minimum Value (multivariable) Conceptual V2

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.
Maximum and Minimum Value (multivariable) Conceptual V3

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Find all local extrema or saddle points for \(f(x,y)=y^3-6y^2-2x^3-6x^2+48x+20.\)

    Saddle Points at \((-4,0,-140)\), \((2,4,44)\)
    Local Minimum at \((-4,4,-172)\)
    Local Maximum at \((2,0,76)\)


    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.
    Maximum and Minimum Value (multivariable) Exercise V1


  2. Find all local extrema or saddle points for \(f(x,y)=x^3+6xy-2y^2.\)

    Saddle Point at \((0,0,0)\)
    Local Maximum at \(\left(-3,-\frac{9}{2},\frac{27}{2}\right)\)
    No Local Minimum


    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.
    Maximum and Minimum Value (multivariable) Exercise V2


  3. A box with no lid is to hold 10 cubic meters. Find the dimensions of the box with a minimum surface area.

    Length\(\; =\sqrt[3]{20}\)
    Width\(\; =\sqrt[3]{20}\)
    Height\(\; =\dfrac{\sqrt[3]{20}}{2}\)


  4. ​Find the absolute extrema of \(f(x,y)=x^2+y^2-2x\) on the closed triangular region with vertices \((2,0)\), \((0,2)\), and \((0,-2).\)

    Absolute Maximum: \(z=4\)
    Absolute Minimum: \(z=-1\)


    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.
    Maximum and Minimum Value (multivariable) Exercise V4


  5. Find the absolute extrema of \(f(x,y)=x^2+y^2-2x\) on the closed circulur region \(x^2+y^2\leq 4.\)

    Absolute Maximum: \(z=8\)
    Absolute Minimim: \(z=-1\)


    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.
    Maximum and Minimum Value (multivariable) Exercise V5


 

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. What is the difference between local extrema and absolute extremam of a surface \(z=f(x,y)\)?
  2. Can a surface \(z=f(x,y)\) have more than one local maximum?
  3. Does a critical point always yield either a local maximum, minimum, or saddle point? How do we classify a critical point as a local maximum, minimum or saddle point?
  4. Under what conditions are we guaranteed a surface will attain both an absolute maximum and an absolute minimum?
  5. Under what conditions does \(z=f(x,y)\) attain an absolute maximum or minimum on the boundary of the closed set \(D\)?
  6. Outline the steps in locating the absolute extrema of \(z=f(x,y)\) on a closed, bounded set \(D\).