Def!n
f-
.
:
has
a
an
✗
for
.
7h1
DERIVATIVES
OF
:
P!lot
f I
s
F rst
der!vat!ve
1ft
has
def!ned
✗a
"
f- +
Know
:
m n
,
'
f-
=
max
then
f-
'
✗→ ✗
f ! Ho )
≤ f-
!nter!or
( ✗ a)
=
-
f- (
0
=
s
Some
Open
!nterval
conta!n!ng
7-
✗ a)
1×0 )
po!nt
≥
"
✗
o
!ts
of
doma!n
and
!f
f
'
f-
≥o
f
0 ≤
'
'
=
≤
o
0
:
!
a
b
↓
ı
local
Nate ! !
vveconcıude
that
(
extrema
•
•
•
:
Cd
e
local
or
po!nts
Where
!nfer!or
po!nts
Where
!nter!or
Of
↓
max
local
m!n
doma!n
the
When
!nfer!or
end po!nts
f
↓
loıal
Noth ng
Max
mn
\
!!.
|
↓
local
f ]
:
'
.
,
.
[a.
:
:
!!.
absolute )
'
f
=
'
f
0
!nterval
( orun on of
nterval )
,
the
Only
p!ace
Where
:
.
s
doma!n off
are
an
s
def!ned
un
.
.
off
'
po!ntoff.pl?F!nd!npabs0lUteextremaotactsfnonaf!n!tecl0sed
D-etn.HN
po!nt
Of
doma n
f
Where
s
Zero
or
Undef!neol
F nd
1.
2
.
i
i
i
i
i
i
i
i
i
3.
alt
cr t cal
Evaluate f
Take
the
po!nts
at
largest
cr!t!cal
and
of
f
po!nts
smallest
n
that
and
of
!nterval
end
those
po!nts
.
.
Values
.
Called
s
!nterval
i
That
0
≤
xö
.
i
such
.
Doma!n
i
Xp
0
0+7×0
l!m
=
✗→
EX
FIXI
l!m
=
'
✗o
at
'
f- + ( ✗ a)
1-1×7
,
atan
or
f
=
!f there
extreme
local
for
test
f)
of
✗o
at
S!m!larly
def!ned
at
max mum
( n doma n
local
a
( relat ve )
10cal
m n
Local
!s
i
i
i
i
APPLICATIONS
rev!ously.wed!scussedabsolu!emaxandm!n.net/t:L0Cdlm!nandmax
i
i
i
i
i
i
i
i
i
i
i
4.
:
a
cr!t!cal
fn
may
have
hlxl
and
✗
lf
2)
of
Max
[ -2,1 ]
on
=
f 'IN
.
1)
[ -2,1 ]
on
213
[ -2,3 ]
on
ZX
=
f!lm
0
=
Observe
211=0
then
tl0 )
that
0
=
✗ =D
→
f-
,
1)
g 'IN
Then
↳
2)
91 2)
3)
50
Abs
=
1)
NIXI
2)
hl
abs
suppose y
fla )
(1)
i
i
i
i
i
i
[-2,1 ]
at
✗
✗
=
-2
1
=
.
cr!t!cal
a
po!nt
h
because
DNE
'
at ✗ = 0
.
absolute Max
→
0
absolute m n
→
flx )
=
!s
-1lb )
s
,
,
If
both
be
4
ahs
0
'
has
number
C
abs
m!n
an
.
( a!s )
0h
la , b)
!n
and
such
abs
.
that
f- (C)
'
( by
Max
=
0
.
EVT )
'
!s
Zero
undet!ned
s
d!ff be
the
max
and
Max
Constant
=
One
le
aıb
t.!s
e!ther
!f
t
Where
t
f
,
d!fferent!al b
at
only
Where
and
least
at
[ aıb ]
po!nts
hypothes!s
there
on
occur
[ aıb ]
on
cts
then
cts
can
f. (C)
at
atta!ned
,
s
✗ =D
→
end po!nts
Now
7
s
,}
!nter!or
(3)
By
¥
!nter!or
(2)
¢
2113
=
atta!ned
-32 ,
01-9
2113
7
=
s
9
✗
=
Theorem)
f
There
i
1
[ 2,3 ]
on
•
Of
Max
.
39T
=
=
s nce
SO
.
34T
=
=
h 10 )
7hm :( Rolle
3-
=
2)
-
m!n
.
(1)
@
✗
because
cons!der th s
-32 ,
✗ 213
h (3)
•
po!nt
Max
-4×3+8=0
0
=
not
=
-
HLXI
!!!.
•
da
we
and
•
=
cr t cal
911×1=-4×3+8
!!.
:
tl1 )
and
=L,
Only
Absolute
m n
±f
the
s
↓
Absolute
lf
0
.
2)
1-
↓
i
i
i
!!!.
:
2
✗
=
m!n
91×1=8×-1/4
.
i
i
i
i
i
i
i
i
i
f- 1×1
.
501h
absolute
F!nd
:
for
fn
,
SO
m!n
O
abs
m!n
12 )
at
s
an
OCLUR
.
any
cf
( aıb )
☐
not
poss!ble
!nter!or
at
end
.
Po!nte
po!nts
,
,
we
then
Know
f. (c)
because
=
0
tl01
=
.
flb )
,
f
Must
i
i
Nate
Statement
!n
Hypotneses
:
essent!al
are
.
O
•
-
:
.__÷
^
µ
not
at
cts
:
•
b
a
|
⑨
*
↓
a
a
b
,
:
:
:
a
:
not
>
b
↓
d.!.ffblehere.tl/:Anappl!cat!0nofR0lle:
not
Ctshere
Shaw
soln
that
FIX )
the equat!on
✗ 3+3×+1
3+3×+1
✗
0
=
has
real
one
exactly
Solut!on
.
notef!scts.tl
Let
:
1)
-
f- (a)
=
-1-3+1
>
1
=
IVT
By
=
-3
=
ex!sts
for
value
suppose
po!nt
contrad!cts
the
0h
!s
( -1,0 )
another
ex!sts
that
f ( ✗ 1) =D
root
✗
z
between
C
Some
fact
:
f 'IN
=
✗ 2=1×1
:
×
.
,
and
✗
3×2-13 ≥
f( ✗ 2)
and
ı
such that
,
3
☐
=
flx )
s
Cts
( aıb )
!n
C
MVT
Theorem
y
f
=
0
.
(Keep
Over
at
closed
!nterval
f- (c)
'
wh!ch
=
[ aıb]
f- (b)
-
and
d!ffble
on
(
a,
b)
.
Then
there
f- (a)
a
ssapeottnel!ne.l!
!sflbj-f-a.la
:
.
:
←
b
b- a
:
/ c)
.
)
Proof
'
!n
M!nd
☆
)
b-
a
0
=
II.
,
Mean
there
,
f-
there
that
rolle thm
Th!s
× , Of
Zero
a
Contrad!ct!on
a
By
İ
< 0
0
there
Assume
and
Let
91×1
=
f( a)
+
(
flbn.tt#)lx-a)andlethlxl=flXl-9lX)--flxl-fla)-(flbl-f(aI)
b- a
÷;Y=f(
×)
Observe
İ
!
I
I
I
I
a
b
nın
9
•
h!s
•
h!sd!ffb.!e
•
hla )
cts
=
on
hlb )
[ aıb ]
onla / b)
=
0
b
(
✗
-
a)
!s
at
least
One
)
ex!sts
there
(c)
'
'
f- (c)
=
(f.
-
(b)
'
such
( alb )
C!n
la )
f-
-
b- a
ftp.t.f#EX-:Cons!derflxl--x2on
f- (c)
)
h 'IN
that
=
0
.
0
=
=
MVT
that
Says
[0,2 ]
there
( 0,2 )
C n
s
f- (c)
'
such that
=
tl2 )
tl0 )
-
=
2- 0
f-
'
IX )
1-
ZX
=
(1)
'
(C)
=
=
21=2
wef!nd
then
[
=
1
42--0-0=2
.
2
.
•
◦
-11tl
lf
:
tl14
:
!
2
1- E
,
lf
f
-
Says
'
[aıb ]
!s
velol!ty.at
=
MVT
☒
!
1
Istanbul
e. g.
pos!t!ve
s
C
450km
there
that
pos!t!on
the
t me
Ankara
fHz)-l
s
the
,
Say
.
Some
(
on
fl (c)
=
a,
,
for
as
same
tr!p
3
s
tl12 )
f-
( ✗ 1)
=
f- ' (C)
=
1- Hz )
-
tl11 ) >
0
.
(
✗ 2- ✗
then
for
Some
C
1) >
average
Veloc!ty
,
✗2
between
0
< flvz)
!s Some
on
( aıb )
C!n
[aıb ]
such that
.
.
such
✗1
=
tl4 )
there
hours
MVT
-
Says that
[ 0,33
C!n
b)
MVT
then
↓
✗ 2- × ,
i
i
i
i
i
i
rolle ,
h
i
i
i
By
that
n
✗1
4÷
veloc!ty.at/-!mec
19lb )
and
w!th
✗2
.
✗
14×2
,
we
have
=
150km/h