Question

If $f(x)$ is differentiable everywhere then

• A. $|f(x)|$ is differentiable everywhere.
• B. $|f|^2$ is differentiable everywhere
• C. $f | f |$ is not differentiable at some point
• D. none of these.

Answer 1

Answer: (B)

If $f(x)$ is differentiable, $\{f(x)\}^2.$ i.e. $|f(x)|^2$ is also differentiable.
Taking $f(x) = x$, we have $|f|(x) = |f(x)| = |x|$, which is not differentiable at $x = 0$.
Thus f is differentiable in $R \ne | f |$ is differentiable in R.
Also $(f | f |)(x) = f (x) | f (x) |= \begin{cases} -\{f(x)\}^2 & \quad \text{if } f(x) < 0\\ \{f(x)\}^2 & \quad \text{if } f(x) \geq 0 \end{cases}$
Which is differentiable in R if f is differentiable in R