Example
Consider the curve
y
=
f
(
x
), where
f
0
(
x
) = 4
x
2
/
5
.
Find the open intervals where
f
is increasing and, decreasing, and where
f
has
relative (local) extrema. Also ﬁnd the intervals where where
f
is concave up and
concave down, and where
f
has point(s) of inﬂection. Then ﬁnd the
y
intercept
and
x
intercept(s) of the curve, and any vertical asymptotes. Determine the
y

coordinates associated with locations of local extrema and point(s) of inﬂection,
if any, and use this information to sketch the graph of the curve.
Solution:
The domain of the function
f
is (
∞
,
∞
). Since
f
0
(
x
) = 4
·
2
5
x

3
/
5
=
8
5
x
3
/
5
it follows that
f
0
(
x
) is undeﬁned at
x
= 0 and that
f
0
(
x
) is never zero. Therefore, there is
only one critical point, at
x
= 0
.
A sign chart for
f
0
follows:
interval
f
0
(
x
) =
8
5
x
3
/
5
behavior of
f
(
∞
,
0)

decreasing
(0
,
∞
)
+
increasing
Thus:
•
f
is increasing on (0
,
∞
),
•
f
is decreasing on (
∞
,
0).
Applying the First Derivative Test to the critical point
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 Fall '10
 KIHYUNHYUN
 Calculus, Derivative, Mathematical analysis, Convex function, Graph of a function

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