Let $\text{⨍ : R→R}$ be a function defined by $⨍ \left(\mathrm{x}\right)=\max \left\{\mathrm{x},{\mathrm{x}}^3\right\}.$ Then set of all points where $⨍ \left(\mathrm{x}\right)$ is not differentiable is

$\mathrm{f}:\mathrm{R}\to \mathrm{R}, \mathrm{f}\left(\mathrm{x}\right)=\max \left\{x,{\mathrm{x}}^3\right\}$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\mathrm{x} ; \left(-\infty , -1\right)\\ {\mathrm{x}}^3 ; \left[-1, 0\right]\\ \mathrm{x} ; \left(0, 1\right)\\ {\mathrm{x}}^3 ; \left(1, \infty \right)\end{array}\right. \\ \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{in} \mathrm{R} \end{gathered}$$

$$\begin{gathered} \mathrm{At} \mathrm{x}=-1 \\ {\mathrm{f}}^1\left(-1^{-}\right)=\underset{\mathrm{x}\to -1^{-}}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(-1\right)}{\mathrm{x}-\left(-1\right)} \\ =\underset{\mathrm{x}\to -1}{\lim }\frac{\mathrm{x}+1}{\mathrm{x}+1}=1 \\ {\mathrm{f}}^1\left(-1^{+}\right)=\underset{\mathrm{x}\to -1}{\lim }\frac{{\mathrm{x}}^3-\left(-1\right)}{\mathrm{x}-\left(-1\right)}=\underset{\mathrm{x}\to -1}{\lim }\left({\mathrm{x}}^2-\mathrm{x}+1\right)=3 \\ {\mathrm{f}}^1\left(-1^{-}\right)\ne {\mathrm{f}}^1\left(-1^{+}\right) \\ \therefore \mathrm{f} \mathrm{is} \mathrm{not} \mathrm{differentiable} \mathrm{at} \mathrm{x}=-1 \end{gathered}$$$$\begin{gathered} \mathrm{Similary} {\mathrm{f}}^1\left(1^{-}\right)=1, {\mathrm{f}}^1\left(1^{+}\right)=3 \\ {\mathrm{f}}^1\left(1^{-}\right)\ne {\mathrm{f}}^1\left(1^{+}\right) \\ \mathrm{f} \mathrm{is} \mathrm{not} \mathrm{differentiable} \mathrm{at} \mathrm{x}=1 \\ \mathrm{f} \mathrm{is} \mathrm{not} \mathrm{differentiable} \mathrm{in} \left\{-1,1\right\} \end{gathered}$$