Graphics Reference
In-Depth Information
z
z
z
y
y
y
x
x
x
f(x,y) = x
2
+ y
2
f(x,y) = -x
2
- y
2
f(x,y) = x
2
- y
2
(a)
(b)
(c)
Figure 4.13.
Extrema for functions of two variables.
An inflection point is a place where the “concavity” of the graph of a function
changes from “concave upward” to “concave downward” or vice versa. The derivative
does
not
have to vanish at an inflection point. The origin is an inflection point of the
function x
3
. See Figure 4.12(b).
Theorem 4.5.3 is not the end of the story for finding extrema for functions of one
variable. One must still check a function on its endpoints. For example, the function
f(x) = x defined on [0,1] does not have any critical points on [0,1] but obviously has
a maximum and a minimum, but these come on the endpoints of the interval.
Next, we want to extend the results about functions of one variable to functions
of two variables. The canonical examples are the functions -x
2
- y
2
and x
2
+ y
2
, which
have a local maximum and a local minimum, respectively, at the origin (see Figure
4.13(a) and (b)), but there is one more possibility. Consider the function
2
2
(
)
=-
fxy
,
x
y
.
(4.13)
See Figure 4.13(c). Although the origin is a critical point for f, it is a minimum along
the x-axis and a maximum along the y-axis, that is, it is not a relative extremum but
a “saddle point.”
Definition.
A critical point of a function that is not a relative extremum is called a
saddle point.
Let
X
Õ
R
2
and let f :
X
Æ
R
be a C
2
4.5.4. Theorem.
function. Assume that
p
is a
critical point for f that lies in the interior of
X
. Let
(
)
2
Dff f
=
-
p
.
xx yy
xy
(1) If D > 0, then
p
is a relative extremum for f that is a relative maximum if
f
xx
(
p
) < 0 and a relative minimum if f
xx
(
p
) > 0.
(2) If D < 0, then
p
is a saddle point.
(3) If D = 0, then nothing can be concluded from this test.
