Find the maxima, minima and point of inflexion of the function
f(x) = 3x^5 - 15x^4 + 25x^3 - 15x^2+ 7.
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Answer:
Inflection points are points of the graph of
f
at which the concavity changes.
In order to investigate concavity, we look at the sign of the second derivative:
f
(
x
)
=
x
4
−
10
x
3
+
24
x
2
+
3
x
+
5
f
'
(
x
)
=
4
x
3
−
30
x
2
+
48
x
+
3
f
(
x
)
=
12
x
2
−
60
x
+
48
=
12
(
x
2
−
5
x
+
4
)
=
12
(
x
−
1
)
(
x
−
4
)
So,
f
'
'
never fails to exist, and
f
'
'
(
x
)
=
0
at
x
=
1
,
4
Consider the intervals:
(
−
∞
,
1
)
,
f
'
'
(
x
)
is positive, so
f
is concave up
(
1
,
4
)
,
f
'
'
(
x
)
is negative, so
f
is concave down
(
4
,
∞
)
,
f
'
'
(
x
)
is positive, so
f
is concave up
The concavity changes at
x
=
1
and
y
=
f
(
1
)
=
23
.
The concavity changed again at
x
=
4
and
y
=
f
(
4
)
=
17
.
The inflection points are:
(
1
,
23
)
and
(
4
,
17
)
.
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