Graphics Reference
In-Depth Information
Then the matrix of partials
∂
∂
f
y
∂
∂
f
z
Ê
ˆ
1
1
Á
Á
Á
˜
˜
˜
zy
yz
Ê
Ë
ˆ
¯
M
=
=
∂
∂
f
y
∂
∂
f
z
-
2
2
2
Ë
¯
is nonsingular at
p
. The set
C
can now be parameterized by the function
()
=
(
()
)
g xxgx
,
,
where g :
A
Æ
R
2
is the function defined on a neighborhood
A
of 1 in
R
guaranteed to
exist by the implicit function theorem.
4.5
Critical Points
In this section we review some basic results about maxima and minima of functions,
in particular, for functions of one or two variables.
4.5.1. Theorem
. Let
X
Õ
R
n
and let f :
X
Æ
R
be a continuous function. If
X
is
compact, then f assumes both a minimum and maximum value on
X
, that is, there
are
p
1
and
p
2
in
X
so that
()
£
()
£
()
f
ppp
f
f
1
2
for all
p
in
X
.
Proof.
By Theorem 4.2.11, the set
Y
= f(
X
) is compact. By Theorem 4.2.4,
Y
is closed
and bounded, so that both inf
Y
and sup
Y
belong to
Y
. Choose any
p
1
and
p
2
in
X
with inf
Y
= f(
p
1
) and sup
Y
= f(
p
2
).
Definition.
Let
X
Õ
R
n
be an open set and let f :
X
Æ
R
be a differentiable function.
Let
p
Œ
X
. If Df(
p
) = 0, then
p
is called a
critical point
of f and f(
p
) is called a
critical
value
.
Note that from a practical point of view, to check whether a point
p
is a critical
point of f one simply checks if all the partials of f vanish at
p
.
Definition.
Let
X
Õ
R
n
and let f :
X
Æ
R
. The function f is said to have a
local
maximum
at a point
p
in
X
if f(
q
) £ f(
p
) for all
q
in a neighborhood of
p
. The func-
tion f is said to have a
local minimum
at
p
if f(
q
) ≥ f(
p
) for all
q
in a neighborhood
of
p
. A point
p
is a
local extremum
if it is either a local maximum or local minimum.
A point
p
in
X
is a (global)
maximum
for f if f(
q
) £ f(
p
) for all
q
in
X
. The point
p
is
a (global)
minimum
for f if f(
q
) ≥ f(
p
) for all
q
in
X
. An
extremum
for f is either a
maximum or minimum for f.
