Absolute maxima and minima of a function on a constraint set (2 variables)

If you cannot load the graph below, try to use another browser. Make sure Java Plugin is enabled and you have allowed Java applets to run.

Alternatively, you may try to do the following:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.
2. Click here to open GeoGebra.
3. After you open GeoGebra, click "File" in the toolbar, then click "Open Webpage...".
4. Enter the web address of this page and click "Open".
5. Now you should be able to view the graph inside GeoGebra.

To rotate the graph, right click and drag.

In this problem, we would like to find the extreme values of \[f(x,y) = 4-\dfrac{1}{4}(x^2+y^2)\] strictly on the constraint set \[g(x,y) = (x-1)^2 + 4y^2 - 4 = 0.\] At each of the critical points shown in the graph below, the level curve of \(f(x,y)\) (the black curve) is tangent to the constraint set (the blue curve). Since \(\nabla f \) is perpendicular to the level curve of \(f\) and \(\nabla g\) is perpendicular to the constraint set, we have \(\nabla f\) parallel to \(\nabla g\) at the critical points, or simply \[\nabla f = \lambda \nabla g\] for some scalar \(\lambda\).

On the constraint set, the maximum value of \(f\) is \(\dfrac{23}{6} = 3.8333\dots\) and the minimum value of \(f\) is \(\dfrac{7}{4} = 1.75\), both attained when the level curve just touches (tangent to) the constraint set.

Joseph Lo, Created with GeoGebra