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


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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:

  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, lagrange2d.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