Home Competitions HMMT 2007 K9
gi s a t w i c e d i f f e r e n t i a b l e f u n c t i o n o v e r t h e p o s i t i v e r e a l s s u c h t h a t is a twice differentiable function over the positive reals such that i s a tw i ce d i f f er e n t iab l e f u n c t i o n o v er t h e p os i t i v er e a l ss u c h t ha t g ( x ) + 2 x 3 g ′ ( x ) + x 4 g ′ ′ ( x ) = 0 for all positive reals x . lim x → ∞ x g ( x ) = 1 \begin{aligned}
g(x)+2 x^{3} g^{\prime}(x)+x^{4} g^{\prime \prime}(x) & =0 \text { for all positive reals } x . \\
\lim _{x \rightarrow \infty} x g(x) & =1
\end{aligned} g ( x ) + 2 x 3 g ′ ( x ) + x 4 g ′′ ( x ) x → ∞ lim xg ( x ) = 0 for all positive reals x . = 1 F i n d t h e r e a l n u m b e r Find the real number F in d t h er e a l n u mb er s u c h t h a t such that s u c h t ha t
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