MATH 2210 Multivariable Calculus Spring 2006
Project #2 Optimization (Section 11.7)
Purpose: This project explores optimization in the multi-variable setting.
NOW WAIT JUST A SECOND! We did optimization on functions and relations involving
more than one variable in Calc I. Correct. The key was having a constraint that allowed
us to reduce the problem to a single variable. We now examine the genuine multi-
variable setting.
Optimization
As in Calc I, we want to identify local maximum and local minimum function values of a
surface (or hyper-surface). It should come as no surprise that we need a way to
determine critical points and test for concavity. (Note: This will involve quite a bit of
algebra – the calculus is fairly easy. But then, so is the algebra…just not as interesting.)
Critical Points – The First Derivatives
Critical points in this context are points (x,y) or (x,y,z) in the domain for which ALL
first partial derivatives are equal to zero (0).
Consider the function
52055),(
44
xyyxyxf
similar to the one on page 813 of
your text used in class. We have
)(20),(
3
yxyxf
x
for which
0
x
f
along the curve
3
xy
.
Likewise, we have
)(20),(
3
xyyxf
y
for which
0
y
f
along the curve
3
yx
.
The critical points (x,y) for this function satisfy
),(0),( yxfyxf
yx
, simultaneously.
That is, the critical points (x,y) occur at the intersection of these two curves.
Problem #1: Find all critical points (x,y) for f.
Local Extrema and Saddle Points – The Second Derivative Test
Once again, as in Calc I, we will use a test for concavity to classify each critical point.
Since the value of the second partial derivatives
xx
f
and
yy
f
indicate the concavity
with respect to either x or y independently, it is clear that these values must have the
same sign if there is any possibility of a local maximum or local minimum. Thus, if
xx
f
and
yy
f
indicate the concavity we must have
0
yyxx
ff
.
However, this alone is not enough information. The mixed partial derivatives
xy
f
and
yx
f
must also be considered. You are reminded that the mixed partial derivatives
measure how much the corresponding tangent plane is “twisting” at a given point. A
