Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two non-constant differentiable functions. If $$f ^ { \prime } ( x ) = \left( e ^ { ( f ( x ) - g ( x ) ) } \right) g ^ { \prime } ( x ) \text { for all } x \in \mathbb { R }$$ and $f ( 1 ) = g ( 2 ) = 1$, then which of the following statement(s) is (are) TRUE? (A) $f ( 2 ) < 1 - \log _ { \mathrm { e } } 2$ (B) $f ( 2 ) > 1 - \log _ { \mathrm { e } } 2$ (C) $g ( 1 ) > 1 - \log _ { \mathrm { e } } 2$ (D) $g ( 1 ) < 1 - \log _ { e } 2$
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two non-constant differentiable functions. If
$$f ^ { \prime } ( x ) = \left( e ^ { ( f ( x ) - g ( x ) ) } \right) g ^ { \prime } ( x ) \text { for all } x \in \mathbb { R }$$
and $f ( 1 ) = g ( 2 ) = 1$, then which of the following statement(s) is (are) TRUE?\\
(A) $f ( 2 ) < 1 - \log _ { \mathrm { e } } 2$\\
(B) $f ( 2 ) > 1 - \log _ { \mathrm { e } } 2$\\
(C) $g ( 1 ) > 1 - \log _ { \mathrm { e } } 2$\\
(D) $g ( 1 ) < 1 - \log _ { e } 2$