The Geometry Behind Lagrange Multipliers

Our model problem will be to find the maximum and minimum values of the function

f(x,y)=x^2+y^2

when x and y are constrained to lie on the ellipse

(x-1)^2+4*y^2=4.

Question 1

Launch Maple and type

with(plots):

at the command line. Then use gradplot to plot the gradient vector field grad f(x,y) for x=-2..4,y=-2..2. Print out a copy of this plot.

What are the level curves for f? On your printout sketch the level curves of height k for several values of k. Make sure you label the heights for each of the curves.

Question 2

Write down a parametrization

alpha(t)=((x(t),y(t))

for the ellipse

(x-1)^2+4*y^2=4

and plot it using the plot command. Then use the display command to simultaneously display the ellipse and grad f.

Think of starting at the point (1,1) on the ellipse and moving a little bit in the clockwise direction around the curve. Based on the direction of grad f at the point (1,1), decide whether f increases or decreases as a result. What if you move counterclockwise? Give reasons for your answers.

Let g(x,y)=(x-1)^2+4*y^2. Then we can think of the ellipse as the level set of height 4 for g. On a printout of your plot from above, sketch grad g at the point (1,1). Is grad g parallel to grad f at this point?

Repeat the above for the point (1,-1).

Question 3

On your plot from the previous question, approximately locate all places where grad f is parallel to grad g. (Note: There are more than two such points!)

Repeat the following for each of the points (x,y) you found.

Evaluate the function f at (x,y).
Does f increase or decrease as you move a slight distance in the clockwise direction from (x,y) along the ellipse? What if you move counterclockwise?
Decide whether the restriction of f to the ellipse has a local maximum or minimum value at (x,y). Give reasons for your answer.

Based on your answers to the above, what are the maximum and minimum values of f along the ellipse g? At what points do these extrema occur? Again, give reasons for your answers.

Question 4

The method of Lagrange mutlipliers says that the extreme values for f when restricted to the level curve g=k will be found among the points (x,y) satisfying

grad f(x,y)=c*(grad g(x,y))

for some constant c. (Each such constant c is called a Lagrange multiplier.) Explain why the points you found in the first part of Question 3 are exactly the points that satisfy the Lagrange multiplier equation.

Next: Verify the Method
Previous: Outline

Robert E. Thurman<thurman@geom.umn.edu>

Document Created: Sat Jan 13 1995
Last modified: Tue Jan 16 11:44:56 1996