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Mathematics
Mathematics
Problem
For each real number
k
k
k
, define
f
k
(
x
)
=
x
3
−
3
x
2
+
k
x
f_k(x)=x^{3}-3x^{2}+kx
f
k
(
x
)
=
x
3
−
3
x
2
+
k
x
on the interval
[
0
,
4
]
[0,4]
[
0
,
4
]
. For which of the following values of
k
k
k
does
f
k
f_k
f
k
have exactly one critical point in
(
0
,
4
)
(0,4)
(
0
,
4
)
, and that point yields the absolute minimum of
f
k
f_k
f
k
on
[
0
,
4
]
[0,4]
[
0
,
4
]
? a)
k
=
3
k=3
k
=
3
b)
k
=
6
k=6
k
=
6
c)
k
=
0
k=0
k
=
0
d)
k
=
1
k=1
k
=
1
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