The curve f(x)=x^3-3x^2-9x+9 has a point of inflexion at
Answers
Answer:
Set the second derivative equal to
0
then solve the equation
6
x
−
6
=
0
.
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x
=
1
Find the points where the second derivative is
0
.
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(
1
,
−
12
)
Split
(
−
∞
,
∞
)
into intervals around the points that could potentially be inflection points.
(
−
∞
,
1
)
∪
(
1
,
∞
)
Substitute a value from the interval
(
−
∞
,
1
)
into the second derivative to determine if it is increasing or decreasing.
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Decreasing on
(
−
∞
,
1
)
since
f
''
(
x
)
<
0
Substitute a value from the interval
(
1
,
∞
)
into the second derivative to determine if it is increasing or decreasing.
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Increasing on
(
1
,
∞
)
since
f
''
(
x
)
>
0
An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is
(
1
,
−
12
)
.
(
1
,
−
12
)
image of graph