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13.7.pdfMaximum and Minimum Values In this
13.7.pdfMaximum and Minimum Values In this
13.7.pdfMaximum and Minimum Values...
13.7.pdfMaximum and Minimum Values In this
Page 6
•
The Second Derivative Test for Relative Extrema
In Examples 1, we were able to determine, by simple algebraic manipulations, whether
f
did or did
not possess a relative extremum at a critical point. For more complicated functions, the following
test may be used.
This test is the analog of the Second Derivative Test for a function of one
variable. Its proof will be omitted.
Theorem 2: The Second Derivative Test for Relative Extreme Values
Suppose that
f
(
x, y
) and its ﬁrst and second partial derivatives are continuous throughout a
disk centered at (
a, b
) and that
f
x
(
a, b
)=
f
y
(
a, b
) = 0 (i.e., (
a, b
) is a critical point of
f
). Let
H
(
x, y
)=
f
xx
(
x, y
)
f
yy
(
x, y
)

[
f
xy
(
x, y
)]
2
.
Then
(a)
f
(
a, b
) is a relative maximum value if
H
(
a, b
)
>
0 and
f
xx
(
a, b
)
<
0.
(b)
f
(
a, b
) is a relative minimum value if
H
(
a, b
)
>
0 and
f
xx
(
a, b
)
>
0.
(c) (
a, b, f
(
a, b
)) is a saddle point if
H
(
a, b
)
<
0.
(d) The test is inconclusive if
H
(
a, b
) = 0. In this case, we must ﬁnd some other way to
determine the behavior of
f
at (
a, b
).
Remark:
The expression
f
xx
f
yy

(
f
xy
)
2
is called the
discriminant
or
Hessian
of
f
.
It is
sometimes easier to remember it in determinant form,
f
xx
f
yy

(
f
xy
)
2
=
f
xx
f
xy
f
xy
f
yy
.
Example 3.
Find the relative maximum and minimum values and saddle point(s) of the function
f
(
x, y
)=
y
3
+3
x
2
y

6
x
2

6
y
2
+ 2.
6
Page 7
Example 4.
Find the relative maximum and minimum values and saddle point(s) of the function
f
(
x, y
)=e
x
cos(
y
).
7
Page 8
Example 5.
Find the relative maximum and minimum values and saddle point(s) of the function
f
(
x, y
)=(
x
2
+
y
2
)e

(
x
2

y
2
)
.
8
Page 9
Example 6.
A delivery company accepts only rectangular boxes the sum of whose length and
girth (perimeter of a crosssection) does not exceed 108 in. Find the dimensions of an acceptable
box of largest volume.
9
Page 10
•
Absolute Extrema on Closed Bounded Regions
Recall that if
f
is a continuous function of a single variable on a closed interval [
a, b
], then the
Extreme Value Theorem guarantees that
f
attains an absolute maximum value and an absolute
minimum value. There is a similar situation for functions of two variables.
Just as a closed interval contains its endpoints, a
closed set
in
R
2
is one that contains all its
boundary points. (A
boundary point
of
D
is a point (
a, b
) such that every disk with center (
a, b
)
contains points in
D
and also points not in
D
). For instance, the disk
D
=
{
(
x, y
):
x
2
+
y
2
≤
1
}
,
which consists of all points on and inside the circle
x
2
+
y
2
= 1, is a closed set because it contains
all of its boundary points (which are the points on the circle
x
2
+
y
2
= 1). But if even one point
on the boundary curve were omitted, the set would not be closed (See Figure 6).
Figure 6: Examples of closed and nonclosed sets.
A
bounded set
in
R
2
is one that is contained within some disk. In other words, it is ﬁnite in extent.
According to the Closed Interval Method for functions of single variable, we found the absolute
extrema of the continuous function
f
on [
a, b
] by evaluating
f
at the critical numbers and at the
endpoints
a
and
b
. In terms of closed and bounded sets, we can state the following counterpart of
the Extreme Value Theorem in two dimensions.
Theorem 3: The Extreme Value Theorem for Functions of Two Variables
If
f
is continuous on a closed, bounded set
D
in the plane, then
f
attains an absolute
maximum value
f
(
a, b
) at some point (
a, b
)
∈
D
and an absolute minimum value
f
(
c, d
) at
some point (
c, d
)
∈
D
.
The following procedure for ﬁnding the extreme values of a function of two variables is the ana
log of the one for ﬁnding the extreme values of a function of one variable by Closed Interval Method.
Procedure for Optimizing
f
on a Closed, Bounded Set
D
(1)
Find the values of
f
at the critical points of
f
in
D
.
(2)
Find the extreme values of
f
on the boundary of
D
.
(3)
The absolute maximum value of
f
and the absolute minimum value of
f
are precisely
the largest and the smallest numbers found in Steps (1) and (2).
10
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