Question

Which of the following function is not differentiable at x=0?

• A. $f\left(x\right)=x^2$
• B. $f\left(x\right)=x+|x|$
• C. $f\left(x\right)=2x$
• D. $noneofthese$

Answer 1

The correct option is B $f\left(x\right)=x+|x|$

$A.f\left(x\right)=x^2$

$f^{\prime }\left(x\right)=2x$

defined for all value of x and continuous

$B.f\left(x\right)=x+\left|x\right|$

$f^{\prime }\left(x\right)=1+\frac{\left|x\right|}{x}$

as $x\to 0^{+}$

$\frac{\left|x\right|}{x}\to \ast 1$

$\underset{x\to 0^{+}}{lim}{f}^{\prime }\left(x\right)=1+1=2$

as $x\to 0^{-}\frac{\left|x\right|}{x}\to -1$

$\underset{x\to 0^{-}}{lim}{f}^{\prime }\left(x\right)=1-1=0$

LHL$\ne$RHL

$\therefore$Not differentiable at $x=0$

$C.f\left(x\right)=x$

$f^{\prime }\left(x\right)=2$