Math 2224 Multivariable Calc–Sec. 12.7: Extreme Values and Saddle Points
I.
Review from 1205
A. Definitions
1. Local Extreme Values (Relative)
a.
A function
f
has a local maximum value
at an interior point
c
of its domain
if
f x
( )
!
f c
( )
for all
x
in an open interval containing
c
.
b.
A function
f
has a local minimum value
at an interior point
c
of its domain
if
f x
( )
!
f c
( )
for all
x
in an open interval containing
c
.
2. Critical Number
a. A critical number
of a function
f
is a number
c
in the domain of
f
such that
!
f
(
c
)
is
zero or
!
f
(
c
)
does not exist (undefined). (A stationary point exists where
!
f
(
x
)
=
0
and a singular point exists where
!
f
(
x
)
is undefined.)
b. If
f
has a local maximum or minimum at
c
, then
c
is a critical number of
f
.
3. A point where the graph of a function has a tangent line and where the concavity
changes is called a point of inflection
.
B. The First Derivative Test for Local Extrema
Let
f
be a continuous function on [
a,b
] and
c
be a critical number in [
a,b
].
1.
If
!
f
(
x
)
"
0
on (
a,c
) and
!
f
(
x
)
"
0
on (
c,b
), then
f
has a local maximum of
f(c)
at
x=c
.
2.
If
!
f
(
x
)
"
0
on (
a,c
) and
!
f
(
x
)
"
0
on (
c,b
), then
f
has a local minimum of
f(c)
at
x=c
.
3. If
!
f
(
x
)
does not change signs at
x=c
, then
f
has no local extrema at
x=c
.
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 Spring '03
 MECothren
 Critical Point, Multivariable Calculus, a,b, Nonbounded

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