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.
Activity 1
Launch Maple and type
with(plots):
at the command line.
Use the command plot3d to display the graph of
f(x,y), and describe this surface. Do not include a
copy of this plot in your writeup.
Use the command 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.
Question 1
What are the level curves for f(x,y)? 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.
What is the relationship between the level curves of f(x,y) and
the gradient field of f(x,y)? Make sure your sketches
reflect this property.
Activity 2
Write down a parametrization
alpha(t)=((x(t),y(t))
for the ellipse
(x-1)^2+4*y^2=4.
Plot this parametric curve using the plot command. Then
use the display command to simultaneously display the
ellipse and your plot of grad f(x,y) from the previous
problem. Print out a copy of this plot.
Question 2
Locate the point (1,1) on your plot. Think of starting at
this point on the ellipse and moving a little bit in the clockwise
direction around the curve. Now recall the geometric significance of
gradient vectors. Based on the direction of the vector grad
f(1,1), decide whether f(x,y) increases or decreases as
you begin to move clockwise. What if you begin to move
counterclockwise? Give reasons for your answers.
Question 3
Let g(x,y)=(x-1)^2+4*y^2. Then another way we can think of
the ellipse is as the level set of height 4 for g(x,y).
Calculate the vector grad g(1,1), and sketch this vector on
the printout of your plot from above.
Compare the vectors grad g(1,1) and grad f(1,1). Are
they parallel?
Repeat the above for the point (1,-1).
Question 4
On your plot from the previous question, approximately locate all
places (x,y) where grad f(x,y) is parallel to
grad g(x,y). (Note: There are more than two such
points!) You will not need to do any new calculations or make any new
plots to answer this question. Use your geometric knowledge of level
sets and gradient vectors.
Question 5
Repeat the following for each of the points (x0,y0) you found
in Question 4.
- Evaluate f(x0,y0).
- Use the direction of the vector grad f(x0,y0) to decide whether
f(x,y) increases or decreases as you move a slight distance in
the clockwise direction from (x0,y0) along the ellipse. What if
you move counterclockwise?
- Decide whether the restriction of f(x,y) to the ellipse
has a local maximum or minimum value at (x0,y0). Give
reasons for your answer.
Based on your answers to the above, what are the maximum and minimum
values of f(x,y) along the ellipse g(x,y)=4? At
what points do these extrema occur?
Explain carefully why we should expect to find solutions to the
restricted max/min problem among the places where grad
g(x,y) is parallel to grad f(x,y).
Question 6
The method of Lagrange mutlipliers says that when (x,y) is
restricted to lie on the level curve g(x,y)=k, the extreme
values for f(x,y) will be found among the points where
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 4 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 1996
Last modified: Mon Jan 20 11:55:08 1997