1
MATH1211/LSweek28/TNK/ /10-11(2)
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1211: Multivariable Calculus
Lecture Summary – Week 28
March 7, 2011
We first went through briefly the solution to
Assignment 3
.
We did a quick revision on the procedure of applying the
Lagrange Multipliers Method
to find
extrema of functions subject to constraints. We note that the methods given in
Theorems 3.1 and 3.2
(p.261 and p.264) are only for finding critical points, and these critical points are candidates that
may
or may not
give local extrema. To complete the solution, we need to find out whether these critical
points actually give local/global minima, or local/global maxima, or not. This last part of the solution
may not be easy.
We considered
Exercise 17
on p.271.
Brief solution
: We wish to maximize
, subject
to constraint
, where
, and
. From
we get
. This, together with the constraint equation give the following
system of equations:
Solving (1) and (2) we obtain
, which is
, which implies either
or
. Since we require
, it then follows
. Similarly, solving (1) and (3) gives
.
Substituting into (4) we get
. Since
(in fact
for all
), (6,6,6) is a valid critical point.
[Note: the checking of
for the points
obtained is an important step in using the
Lagrange Multipliers method.]
(The final checking of (6,6,6) really gives the global maximum of
for
is left as an exercise.)
Alternative solution to Exercise 17
: Since
, by the (geometric mean)
≤
(arithmetic mean)
inequality, we get
, with equality holds iff
. Now that
, it follows that
, with equality holds iff
. Thus
is the unique solution that gives the global maximum of
f
.
However, note that
this solution is not using the Lagrange Multipliers method.
We partially solved