Question

Which of the following function is non-differentiable at $x=0?$

• A. $f\left(x\right)=\cos |x|$
• B. $f\left(x\right)=x|x|$
• C. $f\left(x\right)=|x^3|$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=\cos |x|=\cos x$ which is differentiable at $x=0$

$f\left(x\right)=x|x|=\{\begin{array}{cc}{x}^2,& x\ge 0\\ -x^2,& x<0\end{array}$
$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{cc}2x,& x>0\\ -2x,& x<0\end{array}$
$f^{\prime }\left(0^{+}\right)=f^{\prime }\left(0^{-}\right)=f^{\prime }\left(0\right)$
Hence, $f\left(x\right)$ is differentiable at $x=0$

$f\left(x\right)=|x^3|=\{\begin{array}{cc}{x}^3,& x\ge 0\\ -x^3,& x<0\end{array}$
$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{cc}3x^2,& x>0\\ -3x^2,& x<0\end{array}$
$f^{\prime }\left(0^{+}\right)=f^{\prime }\left(0^{-}\right)=f^{\prime }\left(0\right)$
Hence, $f\left(x\right)$ is differentiable at $x=0$