Section 4.3: Derivatives and the shapes of graphs
(continued)
First Derivative Test
: Suppose that
c
is a critical number
of a continuous function
f
, and suppose that
f
′
(
x
) is defined
for all
x
in a neighborhood of
c
(but not necessarily at
c
itself).
(a) If
f
′
changes from positive to negative at
c
, then
f
has a
local maximum at
c
.
(b) If
f
′
changes from negative to positive at
c
, then
f
has
a local minimum at
c
.
(c) If
f
′
is positive on both sides of
c
or negative on both
sides of
c
, then
f
has no local maximum or local minimum
at
c
.
Proof of (a): Say
f
′
(
x
) > 0 for all
x
in (
c
–
r
,
c
) and
f
′
(
x
) < 0
for all
x
in (
c
,
c
+
r
).
Then, by the increasing/decreasing test,
f
is increasing on [
c
–
r
,
c
], so that
f
(
x
) <
f
(
c
) for all
x
in (
c
–
r
,
c
); and likewise, by the increasing/decreasting test,
f
is
decreasing on [
c
,
c
+
r
], so that
f
(
x
) <
f
(
c
) for all
x
in (
c
,
c
+
r
).
Therefore
f
(
x
) <
f
(
c
) for all
x
≠
c
in (
c
–
r
,
c
+
r
), so
f
(
x
)
≤
f
(
c
) for all
x
in (
c
–
r
,
c
+
r
),
implying that
f
has a local
maximum at
c
.
(The proofs of (b) and (c) are similar.)
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A common alternative to the First Derivative Test,
especially handy when it’s hard to evaluate the derivative
of
f
, is the “testpoint method”:
Suppose that
c
is a critical number of a continuous function
f
, and suppose that we have two testpoints
d
L
and
d
R
with
d
L
<
c
<
d
R
such that
c
is the only critical point of
f
in the
interval [
d
L
,
d
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 Fall '11
 Staff
 Calculus, Critical Point, Derivative, Mathematical analysis, local maximum

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