Absolute extrema of a function on a constraint set (3 variables)


Scroll down to the bottom to view the interactive graph.


Here we shows the extreme value of \[f(x,y,z) = x-y-z\] subject to the constraint \[g(x,y,z) = x^2+y^2+z-2 = 0.\] On the constraint set (the blue surface), the lowest value of \(f(x,y,z)\) one can attain is \(-2.5\). This happens when the level surface \(f(x,y,z) = C\) (the red surface) just touches the constraint set, which is precisely at \(C = -2.5\).

There is no maximum value of \(f\) on the given constraint set.


To view the interactive graph:

  1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.
  2. Download this file, lagrange3d.ggb.
  3. Click here to open GeoGebra.
  4. After you open GeoGebra, click "File" in the toolbar, then click "Open".
  5. Choose the .ggb file you just downloaded and click the "Open" button.
  6. Now you should be able to view the graph inside GeoGebra.

To rotate the graph, right click and drag.

Joseph Lo