Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true? A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$ B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$ C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$ D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$
Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true?\\
A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$\\
B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$\\
C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$\\
D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$