If $f (x)$ is differentiable everywhere, then:

Given: $f(x)$ is differentiable everywhere

$|f|$ is differentiable everywhere

Let $f(x)=x$

$\Rightarrow |f(x)|=|x|$

$\therefore |f|$ is not differentiable everywhere

$|f|^2$ is differentiable everywhere

Let $f(x)=x$

$\Rightarrow |f(x)|^2=x^2$

$f(x)=e^x$

$|f(x)|^2=e^{2x}$

Differentiable everywhere

$f|f|$ is not differentiable at the same points

$f(x)=e^x\Rightarrow |f|=e^x$

$f|f|=e^{2x}\rightarrow$ differentiable everywhere

$\Rightarrow$ This option is wrong

$f+|f|$ is differentiable everywhere

$f(x)=x$, $|f|=|x|$

$f+|f|=\left\{\begin{matrix} 0, x < 0\\ 2x, x\geq 0\end{matrix}\right\}$ not

differentiable at $x=0$

Options (ii), (iii), and (iv) are wrong.

$\Rightarrow$ (ii) is correct.