Question

Let $f\left(x\right)=|\sin x|$. Then which of the following statement(s) is/are true?

• A. f is everywhere differentiable
• B. f is not differentiable at $x=n\pi ,n\in Z$.
• C. f is everywhere continuous but not differentiable at $x=(2n+1)\frac{\pi }{2},n\in Z$
• D. f is continuous everywhere

Answer 1

Answer: (B), (D)

The correct options are
B f is not differentiable at $x=n\pi ,n\in Z$.
D f is continuous everywhere

We have
$f\left(x\right)=\left|\sin x\right|$We know that $\left|x\right|$ and $\sin x$ are continuous everywhere.
Hence, $\left|\sin x\right|$ is continuous everywhere for all $x$.
$|x|$ is non-differentiable at $x=0$.
Therefore, $|\sin x|$ is non-differentiable at $x=0$.
So, $|\sin x|$ is non-differentiable when $\sin x=0$, or $x=n\pi ,n\in Z$.
Hence, $f\left(x\right)$ is continuous everywhere but not differentiable at $x=n\pi ,n\in Z$