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Mathematics
Problem
Determine whether the hypotheses of the Mean Value Theorem are satisfied for the function
f
(
x
)
=
x
3
−
6
x
2
+
9
x
f(x)=x^{3}-6x^{2}+9x
f
(
x
)
=
x
3
−
6
x
2
+
9
x
on the interval
[
0
,
3
]
[0,3]
[
0
,
3
]
. State whether the theorem guarantees a value
c
c
c
in
(
0
,
3
)
(0,3)
(
0
,
3
)
such that
f
′
(
c
)
=
f
(
3
)
−
f
(
0
)
3
−
0
f'(c)=\dfrac{f(3)-f(0)}{3-0}
f
′
(
c
)
=
3
−
0
f
(
3
)
−
f
(
0
)
. Provide a brief justification.
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