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

*Scroll down to the bottom to view the interactive graph.*

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.

**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, lagrange2d.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