Question

Show that the function $f$ defined by $f\left(x\right)=\{\begin{array}{cc}2x+3& \text{if}-3\le x<-2\\ x+1& \text{if}-2\le x<0\\ -x+2& \text{if}0\le x\le 1\end{array}$ is not differentiable at $x=-2$ and $x=0.$

Answer 1

$\begin{array}{c}f\left(x\right)=\{\begin{array}{c}2x+3-3\le x<-2\\ x+1-2\le x<0\\ -x+2if0\le x\le 1\end{array}\\ f^{\prime }\left(0^{+}\right)=-1f^{\prime }\left(0^{-}\right)=1\\ f^{\prime }\left(-2^{+}\right)=1f^{\prime }\left(-2^{-}\right)=2\end{array}$

Function is not differentiable at $x=-2$ and $x=0$