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4
2005 Calculus #4: Smooth function - derivative at 0
2005 Calculus #4: Smooth function - derivative at 0
Source:
April 29, 2013
calculus
function
derivative
real analysis
Problem Statement
Let
f
:
R
→
R
f : \mathbf {R} \to \mathbf {R}
f
:
R
→
R
be a smooth function such that
f
′
(
x
)
2
=
f
(
x
)
f
′
′
(
x
)
f'(x)^2 = f(x) f''(x)
f
′
(
x
)
2
=
f
(
x
)
f
′′
(
x
)
for all
x
x
x
. Suppose
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
f
(
4
)
(
0
)
=
9
f^{(4)} (0) = 9
f
(
4
)
(
0
)
=
9
. Find all possible values of
f
′
(
0
)
f'(0)
f
′
(
0
)
.
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