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Mathematics
Mathematics
Problem
Consider the polynomial
P
(
x
)
=
1
+
b
x
+
a
x
2
P(x)=1+bx+ax^{2}
P
(
x
)
=
1
+
b
x
+
a
x
2
. Find the ordered pair
(
a
,
b
)
(a,b)
(
a
,
b
)
so that
P
P
P
shares the same value, first derivative, and second derivative at
x
=
0
x=0
x
=
0
with the function
F
(
x
)
=
cos
x
F(x)=\cos x
F
(
x
)
=
cos
x
. a)
(
1
2
,
0
)
\left(\dfrac12,0\right)
(
2
1
,
0
)
b)
(
−
1
2
,
1
)
\left(-\dfrac12,1\right)
(
−
2
1
,
1
)
c)
(
−
1
2
,
0
)
\left(-\dfrac12,0\right)
(
−
2
1
,
0
)
d)
(
0
,
−
1
2
)
\left(0,-\dfrac12\right)
(
0
,
−
2
1
)
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