3
5
Optimization of a single variable
2
(
)
0
min or max is achieved
0
min
d
f r
dr
d
2
( )
f r
dr
Optimization of two variables
For a function represented by
f
(
ζ
,
r
),
critical points
,
where a min or max might occur, are given by
.
0
)
,
(
)
,
(
0
)
,
(
)
,
(
r
f
r
f
r
r
f
r
f
r
6
)
(
near
min
relative
has
)]
,
(
[
)
,
(
)
,
(
,
0
)
,
(
If
)
1
2
r
f
r
f
r
f
r
f
r
f
r
rr
How to determine whether the critical
points yield maximum or minimum?
point)
saddle
a
called
is
and
max
nor
min
neither
has
)
,
(
)
,
(
)]
,
(
[
If
)
3
)
,
(
near
max
relative
has
)]
,
(
[
)
,
(
)
,
(
,
0
)
,
(
If
)
2
,
2
2
f
r
f
r
f
r
f
r
f
r
f
r
f
r
f
r
f
rr
r
r
rr
properties
these
of
any
have
could
)
,
(
)
,
(
)]
,
(
[
If
)
4
2
f
r
f
r
f
r
f
rr
r
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
(
)
(
)(
)
(
)
a
a
a
a
a
a
a
a
a
k
m
c
X
F
k
m
k
m
m k
k
m
m
c