Calculus Section 4.3
Page9
4.3
Connecting
′
f
and
′
′
f
with the Graph of
f
First Derivative Test for Local Extrema
Theorem 4
First Derivative Test for Local Extrema
The following test applies to a continuous function
f x
( )
.
At a critical point
c
:
1.
If
′
f
changes sign from positive to negative at
c
⇒
2. If
′
f
changes sign from negative to positive at
c
⇒
3. If
′
f
does not changes sign from positive to negative at
c
⇒
At a left endpoint
a
:
If
′
f
<
0
′
f
>
0
(
)
for
x
>
a
⇒
At a right endpoint
a
:
If
′
f
<
0
′
f
>
0
(
)
for
x
>
a
⇒
Example 1
Using the First Derivative Text
a)
Find the critical points of
f x
( ) =
x
3

12
x

5.
Find the function’s local and absolute
extreme values.
Identify the intervals on which
f
is increasing and decreasing.
Solve Analytically
Verify Graphically
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Page 10
b)
Find the critical points of
f x
( ) =
x
2

3
(
)
e
x
.
Find the function’s local and absolute
extreme values.
Identify the intervals on which
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 Spring '10
 John
 Calculus, Critical Point, Derivative, Mathematical analysis, Convex function, Calculus Section

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