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Mathematics
Mathematics
Problem
Let
f
k
(
x
)
=
x
3
−
3
(
k
+
1
)
x
2
+
3
k
x
+
2
f_{k}(x)=x^{3}-3(k+1)x^{2}+3kx+2
f
k
(
x
)
=
x
3
−
3
(
k
+
1
)
x
2
+
3
k
x
+
2
, where
k
k
k
is a real parameter. (i) Find all critical points of
f
k
(
x
)
f_{k}(x)
f
k
(
x
)
in terms of
k
k
k
. (ii) Determine the set of all real values of
k
k
k
for which
f
k
(
x
)
f_{k}(x)
f
k
(
x
)
is non-decreasing on
(
−
∞
,
0
]
(-\infty,0]
(
−
∞
,
0
]
and strictly decreasing on
[
0
,
∞
)
[0,\infty)
[
0
,
∞
)
.
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