$f(x) = \large \ln \left( |e^{\left [ \;\; |{\color{blue}{\sin}} \left ( {\color{red}{|x|}} \right ) | \;\; \right ]}| \right )$
$1. \qquad {\color{red}{\bf |x|}}\rightarrow \;\; f(x)\;\;\; \text{is Even }\rightarrow \quad \text{option b not possible}$
$2. \qquad m = |\;{\color{blue}{\sin}} \left ( {\color{red}{\bf |x|}} \right ) | \;\; \geq \;\; 0 \quad \rightarrow {\color{blue}{e^{\bf m} \;\; \geq \;\; 1}}$
$3.\Rightarrow f(x) = \ln\left ( | \color{blue}{e^{\bf m}}| \right ) = \ln\left ( \color{blue}{e^{\bf m}}\right ) \geq 0 \quad \left \{ \text{from 2} \right\} \;\; \rightarrow \text{option a not possible}$
$4. \text{ and } f(x)_{x=0} = 0 \;\; \rightarrow \text{option d not possible}$
$\Rightarrow \text{answer C}$