Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then (A) $f$ has a local minimum at $x = 2$ (B) $f$ has a local maximum at $x = 2$ (C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$ (D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$
Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then\\
(A) $f$ has a local minimum at $x = 2$\\
(B) $f$ has a local maximum at $x = 2$\\
(C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$\\
(D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$