Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

224_Functions of Several Variables_[Ch. 6
The question of the existence and character of a conditional extremum is
solved on the basis of a study of the sign of the second differential of the
Lagrange function:
yi
dx 2
-
dxdy
for the given system of values of x, y, h obtained from (2) or the condition
that dx and dy are related by the equation
Namely, the function / (x t y) has a conditional maximum, if d*F 0. As a particular case, if the discriminant A
of the function F (x, y) at a stationary point is positive, then at this point
there is a conditional maximum of the function / (x, y), if A < (or C < 0),
and a conditional minimum, if A > (or C > 0)
In similar fashion we find the conditional extremum of a function of
three or more variables provided there is one or several coupling equations
(the number of which, however, must be less than the number of the variables)
Here, we have to introduce into the Lagrange function as many undetermined
multipliers factors as there are coupling equations.
Example 2. Find the extremum of the function
z=:6 4* 3y
provided the variables x and y satisfy the equation
x*-\-y*=\
Solution. Geometrically, the problem reduces to finding the greatest and
least values of the e-coordinate of the plane z 6 4.v 3y for points of its
intersection with the cylinder ji 2
-f// 2 =l
We form the Lagrange function
F(x, y)--=6 4x-3f/-l-M* 2 2
-|-{/
***following
system of We have T = ~ - 4 + 2>jr, = 3 + 2X#. The necessary conditions yield the
equations:
Solving this system we find
and
Since
it follows that
i
:
i - 5 _ 4
X
^-"2"' '-~5~'
____^_ ___
2 ~"
2 1
dx 2 -*"'
^~""5"
=0,
dxdy
f
y
dy 2
1).