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ELATIVE
E
XTREMA
: F
IRST
AND
S
ECOND
D
ERIVATIVE
T
ESTS
Theorem
:
Suppose that f is a function defined on an open interval containing the number
0
x
.
If
f
has a relative extremum at
0
x
x
=
, then either
0
)
(
'
0
=
x
f
or
f
is not differentiable at
0
x
.
values of
x
where either
0
)
(
'
=
x
f
or
)
(
'
x
f
does not exist are called critical values
when
0
)
(
'
=
x
f
, we call these points stationary values (critical points)
note: a critical point does not necessarily imply a relative extremum
Consider the following graphs:
)
(
'
x
f
is positive on the left,
)
(
'
x
f
is negative on the left,
same sign on both sides,
negative on the right
positive on the right
no relative extrema
This leads us to a theorem
)
(
'
x
f
changes signs at relative extrema.
First Derivative Test:
Suppose
f
is continuous at a critical number
0
x
.
Absolute maximum
Relative maximum
Relative minimum
There is an open interval
0
x
on which
)
(
0
x
f
is the
largest value.
The relative maximum and
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 Fall '08
 Kim
 Calculus, Derivative

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