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Mathematics
Mathematics
Problem
A function
h
h
h
is analytic at
x
=
1
x=1
x
=
1
. Its third–degree Taylor polynomial about
x
=
1
x=1
x
=
1
is
P
3
(
x
)
=
9
+
4
(
x
−
1
)
+
k
(
x
−
1
)
2
−
4
(
x
−
1
)
3
,
P_3(x)=9+4(x-1)+k(x-1)^2-4(x-1)^3,
P
3
(
x
)
=
9
+
4
(
x
−
1
)
+
k
(
x
−
1
)
2
−
4
(
x
−
1
)
3
,
where
k
k
k
is a real constant. (a) Given that
h
′
′
(
1
)
=
12
h''(1)=12
h
′′
(
1
)
=
12
, determine the value of
k
k
k
. (b) Find
h
′
′
′
(
1
)
h'''(1)
h
′′′
(
1
)
.
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