Which of the following functions is differentiable at \( x = 0 \)?
Show Hint
Differentiability and Absolute Values}
If \( f(x) \) involves \( |x| \), check for corner points at \( x = 0 \)
Constant functions are always differentiable
Visualize or test left-hand and right-hand derivatives for piecewise cases
\[
f(x) = \sin(|x| - |x|) = \sin(0) = 0 \Rightarrow f(x) = 0 \text{ for all } x
\]
\[
\Rightarrow f(x) \text{ is a constant function } \Rightarrow \text{Differentiable everywhere}
\]
Other functions involve absolute values inside trigonometric functions and result in a non-smooth corner at \( x = 0 \), which causes non-differentiability.