Sections 4.1, 4.2, 4.3, and 4.4: Review Theory
Instructor: Ms. Hoa Nguyen
([email protected])
4.1
Critical Numbers
To ﬁnd critical numbers of a function
f
:
 1. Find the derivative
f
0
(
x
).
 2. Find the critical numbers
in the domain of
f
by considering 2 cases:
f
0
(
x
) = 0 and
f
0
(
x
)
does not exist.
The Closed Interval Method
To ﬁnd the ABSOLUTE maximum and minimum values of a continuous function f on a
closed
interval
[
a,b
], do the following steps:
 1. Find the critical numbers
in
(
) and their fvalues.
 2. Find the values of
f
at the endpoints
of the interval.
 3. Compare the fvalues: The largest value from steps 1 and 2 is the absolute maximum value.
The smallest is the absolute minimum value.
Local Maximum or Minimum
Given a graph of a function
f
on its domain
D
, to know if
f
has a LOCAL max or min at
x
=
c
,
ﬁrst we pick an open interval
containing
c
and
lying inside the domain
D
.
 If we
cannot
pick such an interval,
f
does
not
have local max or min at
x
=
c
. This means
that
f
NEVER has local max or min at the endpoints of the domain
D
.
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 Fall '08
 Noohi
 Calculus, Derivative, open interval

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