Answer:
The conclusion of the Mean Value Theorem says that there is a number
c
in the interval
(
1
,
3
)
such that:
f
'
(
c
)
=
f
(
3
)
−
f
(
1
)
3
−
1
To find (or try to find)
c
, set up this equation and solve for
c
.
If there's more than one
c
make sure you get the one (or more) in the interval
(
1
,
3
)
.
For
f
(
x
)
−
−
2
x
2
−
x
+
2
, we have
f
(
1
)
=
−
1
, and
f
(
3
)
=
−
18
−
3
+
2
=
−
19
Also,
f
'
(
x
)
=
−
4
x
−
1
.
So the
c
we're looking for satisfies:
f
'
(
c
)
=
−
4
c
−
1
=
f
(
3
)
−
f
(
1
)
3
−
1
=
−
19
−
−
1
3
−
1
=
−
18
2
=
−
9
So we need
−
4
c
−
1
=
−
9
. And
c
=
2
.
Note:
I hope you've been told that actually finding the value of
c
is not a part of the Mean Value Theorem.
The additional question"find the value of
c
" is intended as a review of your ability to solve equations. For most functions, you will not be able to find the
c
that the MVT guarantees us is there..