Lagrange Multipliers (Two Variables)

(see below for directions - read them while the applet loads!)

Directions:

• Wait for the applet to load and for the contour plot to appear in the gray area above.
  (If there is no gray area, check your browser settings to make sure that Java is enabled, or try with another browser)

• Setup. Enter the function to minimize / maximize, f(x,y), into the box in the upper-left corner. Enter the constraint, g(x,y), into the box immediately below. Click on the "Plot curves" button in the lower-left corner to update the display. Then, use the yellow slider control to set the value of b in the constraint equation g(x,y)=b.

• The applet shows a contour plot of f (in blue), together with the level curve g(x,y)=b corresponding to the constraint equation (in yellow). You can use the blue slider control to move a highlighted level curve of f. The minima and maxima of f subject to the constraint correspond to the points where this level curve becomes tangent to the yellow curve g(x,y)=b.

• Click in the contour plot to move the pink dot and display the gradient vectors of f and g at the given point. The components of grad(f) and grad(g) are displayed in the lower-right corner. As expected, the two gradient vectors are proportional to each other at a constrained minimum/maximum.

• The red "Show solutions" button displays a red curve consisting of all points where grad(f) and grad(g) are proportional to each other. The Lagrange multiplier method tells us that constrained minima/maxima occur when this proportionality condition and the constraint equation are both satisfied: this corresponds to the points where the red and yellow curves intersect.