Mathematics - Continuity - Differentiability Question with Solution | TestHub

The function $f(x)$ can be written piecewise. By comparing $x,x^2,x^3$, we find $f(x)=x$ for $x<0$, $f(x)=x^2$ for $0\le x<1$, and $f(x)=x^3$ for $x\ge 1$. At $x=0$, $f^{\prime }(0^{-})=1$ and $f^{\prime }(0^{+})=0$, so $f(x)$ is not differentiable at $x=0$. At $x=1$, $f^{\prime }(1^{-})=2$ and $f^{\prime }(1^{+})=3$, so $f(x)$ is not differentiable at $x=1$. Thus, there are 2 points where $f(x)$ is not differentiable.