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MTH 173 setion 4.3 notes, page 1
Definition
A function
is
on an interval if for any two numbers
and
0
B
B
increasing
"
#
in the interval,
implies
.
B B
0 B
0 B
"
#
"
#
a
b
a
b
A function
is
on an interval if for any two numbers
and
0
B
B
decreasing
"
#
in the interval,
implies
.
B B
0 B
0 B
"
#
"
#
a
b
a
b
Thm
Let
be a function that is continuous on the closed interval
and
0
+ß ,
c
d
differentiable on the open interval
.
a
b
+ß ,
"Þ
0 B !
B
+ß ,
0
+ß ,
If
for all
in
, then
is increasing on
.
w
a b
a
b
c
d
2
If
for all
in
, then
is decreasing on
.
Þ
0 B !
B
+ß ,
0
+ß ,
w
a b
a
b
c
d
3
If
for all
in
, then
is constant on
.
Þ
0 B o !
B
+ß ,
0
+ß ,
w
a b
a
b
c
d
Proof:
Let
be in the interval
such that
.
B ß B
+ß ,
+ Ÿ B B Ÿ ,
"
#
"
#
a
b
By the MVT, there is a
where
such that

B  B
"
#
0  o
w
0 B 0 B
B B
a b
a
b
a
b
#
"
#
"
.
By hypothesis,
0  !Þ
w
a b
Since
we know
.
B B ß
B B !
"
#
#
"
Thus in
, the LHS and the denominator on the RHS
0  o
0 B
0 B
B B
w
#
"
#
"
a b
a
b
a
b
are positive. Then the numerator
must also be positive.
0 B
0 B
a
b
a
b
#
"
Thus
0 B
0 B
! Ê 0 B
0 B Þ
a
b
a
b
a
b
a
b
#
"
"
#
Since
and
are any two numbers in
B
B
"
#
a
b
+ß ,
, we know the function increases everywhere in the interval.
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View Full DocumentMTH 173 setion 4.3 notes, page 2
Thm
Let
be a critical number of a function
on an open interval
containing
. If

0
M

0
is differentiable on the interval, except possibly at
, then
can be classified

0 
a b
as follows:
1.
If
changes from negative to positive at
then
is a
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 Spring '09
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