## 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:**

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

To rotate the graph, **right click and drag**.

Joseph Lo