MATH 261, Spring 2001 Due Date: Name(s):
Project 15.7a*: Function Optimization
Objective
The objective of this project is to use Maple to find the extrema of a function of two variables.
Narrative
If you have not already done so, read Section 15.7 in the text.
In this project you will be asked to use Maple to find the extrema of f(x, y) = (3x + 4y)/(x2
+ y2
+ 1).
To do this, you will need to use the commands:
f1 := D[1](f) to define the function f1 as f/x,
f2 := D[2](f) to define the function f2 as f/y,
f11 := D[1,1](f) to define the function f11 as 2
f/x2
,
f12 := D[1,2](f) to define the function f12 as 2
f/xy, and
f22 := D[2,2](f) to define the function f22 as 2
f/y2
,
as well as the command H := (x,y) -> f11(x,y)*f22(x,y)-f12(x,y)^2 to define the Hessian of f(x, y).
You will also need to use the solve command: solve({g(x,y)=0,h(x,y)=0},{x,y}) solves the system of
equations
g(x, y) = 0, h(x, y) = 0
for x and y.
Task
If f(x, y) = (3x + 4y)/(x2
+ y2
+ 1), use Maple to:
a) find the critical points of f, and
b) apply the second derivative test for functions of two variables to classify the values of f at these critical
points as relative maxima, relative minima, or neither. (Your report should contain a statement -- or
statements -- of the form, "f( something, something ) is a ." where ( something, something ) are the
coordinates of a critical point, and you fill in the blank with "relative maximum", "relative minimum", or
"saddle value".)
Finally, ..
c) Draw the graph of f, using an appropriate window size and point of view to confirm your findings.