Question

Let $f\left(x\right)=|\sin x|$, then $f\left(x\right)$ is?

• A. continuous every where
• B. non-differentiable at odd and even multiple of $\pi$
• C. everywhere continuous but non-differentiable at $x=n\pi ,n\in I$
• D. All of these

Answer 1

Answer: (D)

All of these

$f\left(x\right)=\left|\sin x\right|$

Since $\sin x$ is a continuous function $\left|\sin x\right|$ is also a continuous function over R

$f^{\prime }\left(x\right)=\frac{\left|\sin x\right|}{\sin x}\times \cos x(\frac{d}{dx}\left|x\right|=\frac{\left|x\right|}{x})$

$\underset{x\to n{\pi }^{+}}{lim}\frac{\left|\sin x\right|}{\sin x}\times \cos x=\underset{x\to n{\pi }^{+}}{lim}\frac{\sin x}{\sin x}\times \cos x$

$=-\underset{x\to n{\pi }^{+}}{lim}\cos x$

$\underset{x\to n{\pi }^{-}}{lim}\frac{\left|\sin x\right|}{\sin x}\times \cos x=\underset{x\to n{\pi }^{-}}{lim}\frac{\sin x}{\sin x}\times \cos x$

$=\underset{x\to n{\pi }^{-}}{lim}\cos x$

Since Right limit is not equal to left limit

Limit at $n\pi$ does not exist (where n is any integer).