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Extremum & Inflection
Finding and Confirming the Points of Extremum & Inflection
Extremum Inflection
ICritical Points
Points at which the first derivative is equal 0 or does not exist.
Candidate Points for InflectionPoints at which the second derivative is equal 0 or does not exist.
IIThe First Derivative Test for Local ExtremumTesting the sign of the first derivative about the point
The second Derivative Test for inflectionTesting the sign of the second derivative about the point
IIIThe Second Derivative Test for Local ExtremumTesting the sign of the second derivative at the point
The Third Derivative Test for Inflection
Testing the sign of the third derivative at the point
Extremum
I
II.
Critical PointsWe find critical points, which include any point ( x0 , f((x0) ) of f
at which either the derivative f’ (x0) equal 0 or does not
exist.
The First Derivative Test for Local ExtremumFor each critical point ( x0 , f((x0) ) we examine the sign of the
first derivative f’ (x) on the immediate left and the immediate right of this point x0.
If there is a change of sign at x0, then the point ( x0 , f((x0) ) is
a point of local extremum, with the extremum being:
(1) A local maximum if the sign of f’ (x) is positive on the immediate left and negative on the immediate right of the critical point x0
(2) A local minimum if the sign of f’ (x) is negative on the immediate left and positive on the immediate right of the critical point
Extremum
III.The Second Derivative Test for Local Extremum
Only for critical points ( x0 , f((x0) ) for which the
first derivative f’ (x0) = 0
(1) If f’’ (x0) is negative, then the point ( x0 , f(x0) )
is a point of local maximum
(2) If f’’ (x0) is positive, then the point ( x0 , f(x0) )
is a point of local minimum
Inflection
I
II.
Candidate PointsWe find candidate points, which include any point (x0,y0) of f
at which either the second derivative f’’ (x0) equal 0 or does
not exist
The Second Derivative Test for InflectionFor each candidate point ( x0 , f((x0) ) we examine the sign of the
second derivative f’’ (x) on the immediate left and the immediate right of this point x0.
If there is a change of sign at x0, then the point ( x0 , f((x0) ) is a point
of inflection, at which:
(1) The graph of f changes from concave upward to concave downward if the sign of second derivative f’’ (x) changes from positive on the immediate left to negative on the immediate right of the candidate point x0
(2) The graph of f changes from concave downward to concave upward if the sign of second derivative f’’ (x) changes from negative on the immediate left to positive on the immediate right of the candidate point x0
Inflection
III.The Third Derivative Test for InflectionOnly for candidate points ( x0 , f((x0) ) for which
the second derivative f’’ (x0) = 0
(1) If f’’’ (x0) is negative, then the point
( x0 , f((x0) ) is a point of inflection at which the
graph of f changes from concave upward to concave downward
(2) If f’’ (x0) is positive, , then the point
( x0 , f((x0) ) is a point of inflection at which the
graph of f changes from concave downward to concave upward
One Page Summary
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Example
Let:
f(x) = 2x3 – 9x2 +12x + 1
Determine all points of extremum and inflection of the
function f and use this information and other information to
sketch its graph.
Solution
mumilocaloftpoinaisf
imumlocaloftpoinaisf
extremunlocalfortestderivativeondThe
ff
toequalisxfwhichattspointspoinCritical
xxxf
xxxxxxxf
xxxxf
haveWe
min)5,2(0)2
32(12)2(
max)6,1(0)2
31(12)1(
:sec
)5,2())2(,2(&)6,1())1(,1(
0)(:
)2
3(121812)(
)2)(1(6)23(612186)(
11292)(
:
22
23
upwardconcavetodownwardconcavefrom
changesfofgraphthewhichatlectionoftpoinais
f
lectionfortestderivativethirdThe
f
toequalisxfwhichattspointspoinCandidate
xf
xxxf
inf)2
11,2
3(
012)2
3(
:inf
)2
11,2
3())
2
3(,
2
3(
0)(:
2)(
)2
3(121812)(
Graphing f
)2
11,2
3(inf.5
)5,2(min.4
)6,1(max.3
)1,0(int.2
.1
:
?)()(
)1,0())0(,0(:int
11292)(
:
lim
23
atlectionanhasf
atlocalahasf
atlocalahasf
ataxisythecsersef
boundnowithdecreasesxwhen
boundnowithdecreasesandboundnowith
increasesxwhenboundnowithincreasesf
haveweNow
Whyxf
faxisythewithfofctionerseThe
xxxxf
haveWe
x
Graph of f
f’
RonupwardconcaveisgRinxallforxgceSin
mumilocaloftpoinaisg
extremunlocalfortestderivativeondThe
g
toequalisxgwhichattspointspoinCritical
xg
xxxg
xxxx
xxxfxg
Let
,)(
min)2
3,2
3(012)
2
3(
:sec
)2
3,2
3())
2
3(,
2
3(
0)(:
12)(
)2
3(121812)(
)2)(1(6)23(6
12186)()(2
2
Graphing g=f’
Ronupwardconcaveisg
atlocalahasg
ataxisythecserseg
boundnowith
decreasesorincreasesxwhenboundnowithincreasesg
haveweNow
Whyxg
gaxisythewithfofctionerseThe
xxxxxg
haveWe
x
.4
)2
3,2
3(min.3
)12,0(int.2
.
.1
:
?)()(
)12,0())0(,0(:int
)2)(1(612186)(
:
lim
2
Graph of g=f’
Question?Home Quiz (1)
Can you graph g without using the derivatives?
f’’
)0,2
3(&)18,0(
)2
3(12
1812)()(
throughlinestraigttheisThis
x
xxfxh
Let
f’’’
)12,0(
12)()(
throughlinehorizontaltheisThis
xfxv
Let
The relation between f and f’
minint))1(,1(
2
2)(.
maxint))1(,1(
1
1)(.
int))2(,2())1(,1(
)1()1(.3
)2,1(
)2,1(.2
),2()1,(
),2()1,(.1
localofpoaisf
xofrightimmediatethetopositiveand
xofleftimmediatethetonegativeisxfb
localofpoaisf
xofrightimmediatethetonegativeand
xofleftimmediatethetopositiveisxfa
fforspocriticalarefandf
zeroarefandf
ingdecreasisfOn
negativeisfOn
ingincreasisfandOn
positiveisfandOn
The relation between f and f’’
.
intinf))2
3(,
2
3(
2
32
3)(.4
.infint))2
3(,
2
3(
)2
3(.3
),2
3(
),2
3(.2
),2
3(
)2
3,(.1
upwardconcavetodownwardconcavefromchangesfof
graphthewhichatfforpolectionanf
xofrightimmediatethetopositiveand
xofleftimmediatethetonegativeisxf
fforlectionforpocandidateisf
zeroisf
upwardconcaveisfOn
positiveisfOn
downwardconcaveisfOn
negativeisfOn
The relation between f and f’’’
upwardconcaveto
downwarconcavefromchangesfofgraphthe
whichatctionleofpoaisf
f
ctionleofpocandidateaisf
infint))2
3(,
2
3(
0)2
3(
&
infint))2
3(,
2
3(
The Four Graphs graphs
Homework (1)
12)(.2
52)(.1
infint
42
24
xxxf
xxxf
itgraph
andfoflectionandextremumofspoallFind
Homework (2)
For each of the functions f of the previous homework (1)
a. Determine the intervals on which f is increasing or decreasing.
b. Determine the intervals on which f is concave upward or concave downward.