MATH 261, Spring 2001 Due Date: Name(s):
Project 15.8a: Constrained Optimization
Objective
The objective of this project is to illustrate some of the concepts behind constrained function optimization
using Lagrange multipliers.
Narrative
If you have not already done so, read Section 15.8 in the text. In this section we learn that finding the
extrema of the function z = f(x, y) subject to the constraint g(x, y) = 0, can be reduced to solving the
system of 3 equations
f(x, y) = g(x, y), g(x, y) = 0
in the 3 unknowns x, y, and . In this project we consider the specific problem of finding the extrema of
f(x, y) = 3x2
- xy + y2
- 3x - 5y + 9
subject to the constraint (along the curve C whose equation is)
x2
+ y2
= 1.
Task
a) Type the following lines into Maple.
> # Project 15.8a: Constrained Optimization
> restart: with(plots):
> plot0 := contourplot(3*x^2-x*y+y^2-3*x-5*y+9,x=-2..2,y=-2..2,color=red,contours=8):
> plot1 := implicitplot(x^2+y^2=1,x=-2..2,y=-2..2,color=blue):
> display({plot0,plot1});
At this time, make a hard-copy of your typed input and Maple's responses. Then, ...
b) By hand, carefully draw the level curves of f that correspond to the extreme values of f along C. Label
each of the curves you have drawn by hand with the value of f along it. (To find the value of f along each
curve, first estimate the coordinates of the point of intersection of each level curve with C, and f; all you
need is an estimate: you do not have to solve the Lagrange multiplier system of equations! Then use Maple
to substitute the coordinates of each point of intersection into f.)
c) Label the level curves on each side of each of the level curves you have drawn by hand with the value of
f along it. (Use the same technique you used in part (b) to obtain these values.)
d) At each of the points where C meets these two level curves, draw and label the tangent line to C and a
vector in the direction of the gradient f of f. (Remember: The direction of the gradient is the direction
of steepest ascent.)
Comments
There are often ways to approach constrained optimization problems other than Lagrange multipliers. For
example, the problem addressed above can be reduced to a problem in the optimization of a single function
of a single variable by observing that if we parametrize the curve x2
+ y2
= 1 by x(t) = cos t, y(t) = sin t, t 
[0, 2] then, along the graph of g(x, y) = 0,
f = f(t) = cos2
t - cos t sin t + 3 sin2
t - 5 cos t - 3 sin t + 9.