Jessica Chung & Derek Hui
Week 12
QMA PASS  2006 S2  Tues, 34pm  QUAD G042
Questions? Email us: [email protected]
, [email protected]
1/4
M
AXIMA AND
M
INIMA OF
F
UNCTIONS
Most of you would have covered this in high school, if you haven’t let us know!!
There are two distinct types of maxima and minima (just some funky terms to remember), a global/absolute maximum/minimum and a local/relative
maximum/minimum:
•
Given
)
(
x
f
y
=
and
a
x
=
is a point in the domain of
)
(
x
f
•
Global or absolute maximum
occurs if
)
(
)
(
x
f
a
f
≥
for
all
x
in the domain
of
)
(
x
f
•
Global or absolute minimum
occurs if
)
(
)
(
x
f
a
f
≤
for
all
x
in the domain
of
)
(
x
f
•
Local or relative maximum
occurs if
)
(
)
(
x
f
a
f
≥
for
all
x
in some interval
of
)
(
x
f
•
Local or relative minimum
occurs if
)
(
)
(
x
f
a
f
≤
for
all
x
in some interval
of
)
(
x
f
However this is best illustrated visually:
NOTE:
An
extremum
is a term used for
both
maximum and minimum points (comes from “extremity”). Hence, a global or absolute extremum is always
a local or relative extremum as well.
F
INDING
E
XTREMA
First Derivative Test:
•
Find Stationary Points:
o
Solve
0
)
(
'
=
x
f
or
0
=
dx
dy
to obtain
stationary points
•
Check if stationary points are maxima or minima:
o
“Pidgeon hole” test
o
See if
)
(
x
f
is increasing
[
]
0
)
(
'
>
x
f
or decreasing
[
]
0
)
(
'
<
x
f
on either side of the stationary point
o
If
)
(
'
x
f
goes from positive to negative as
x
increases it is a
local maximum
o
If
)
(
'
x
f
goes from negative to positive as
x
increases it is a
local minimum
o
If
)
(
'
x
f
doesn’t change – may be a
point of inflection
(sometimes spelt “inflexion”)
Second Derivative Test:
•
Find Stationary Points:
o
Solve
0
)
(
'
=
x
f
or
0
=
dx
dy
to obtain
stationary points (
turning points
)
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Jessica Chung & Derek Hui
Week 12
QMA PASS  2006 S2  Tues, 34pm  QUAD G042
Questions? Email us: [email protected]
, [email protected]
2/4
•
Calculate the second derivative:
)
(
'
'
x
f
or
2
2
dx
y
d
at each stationary point:
o
If
0
)
(
'
'
<
x
f
then it is a
local maximum
o
If
0
)
(
'
'
>
x
f
then it is a
local minimum
o
If
0
)
(
'
'
=
x
f
then we must check if it is a
point of inflection
Points of Inflection:
•
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 Three '08
 julia
 Fermat's theorem, Jessica Chung, Derek Hui

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