MATH 261, Spring 2001 Due Date: Name(s):
Extra Project 15.8d: Lagrange Multipliers
Objective
The objective of this project is to illustrate how you can use Maple to check your work and answers to
your Lagrange Multipliers homework.
Narrative
If you have not already done so, read Section 15.8 in the text.
One of the toughest aspects of Lagrange Multipliers involves solving the systems of equations that the
technique leads to. In this project we illustrate how you can use Maple to check your work and answers to your
Lagrange Multiplier homework. We work with a specific example: To find the extrema of f(x, y) = x2
+ y2
subject to the constraint (along the curve C whose equation is) x4
+ y4
= 1, we must solve the equation
f = g or 2x, 2y =  4x3
, 4y2
assuming x4
+ y4
= 1. That is, we must solve the system of 3 nonlinear equations in 3 unknowns:
2x = 4x3
, 2y = 4y3
, x4
+ y4
= 1.
Task
1. a) Type the following command lines into Maple in the order in which they are listed. They illustrate
how Maple can be used to solve the above system of equations.
> # Project 15.8d: Lagrange Multipliers
> restart:
> f := (x,y) -> x^2+y^2;
> g := (x,y) -> x^4+y^4-1;
> eq1 := diff(f(x,y),x)=lambda*diff(g(x,y),x);
> eq2 := diff(f(x,y),y)=lambda*diff(g(x,y),y);
> eq3 := g(x,y)=0;
> solution := solve({eq1,eq2,eq3},{x,y,lambda}); (See the discussion below.)
> f(1,0);
> f(-1,0);
> allvalues(solution[3]);
Some observations: The first few lines of code define f and g as well as the equations to be solved. In
this example we are using only one constraint for demonstration purposes, but our code would be easy
to modify if we were using two or more constraints. Then we ask Maple to solve our system; this is
where the magic occurs. It would be nice if we could ask Maple to show its work, but unfortunately that
is not possible. But we should be happy we're at least getting the answer: while this problem might
have looked a little innocuous at the beginning, it has 8 solutions! In typing f(1,0); and f(-1,0);
we're checking the values of f at the first two solutions to determine what the extrema of f are. When
we get to the third solution we find that unknowns are expressed as solutions to equations; by typing
allvalues(solution[3]); we express all the unknowns of the third solution numerically so we can
evaluate f at the third solution. Since we find, however, that one of the unknowns is imaginary (Maple
uses I to denote

-1), we do not need to evaluate f at the third solution.
b) Evaluate f at the relaining five solutions, and state and classify (on the basis of all the information
you have obtained) the absolute maximum and absolute minimum values of f.
2. Do Exercises 3-13, 15-19 on page 990 of the text using Maple.