Question

The number of points at which the function $f\left(x\right)=|2x+1|–3|x+2|+|x^2+x–2|,x\in R$ is not differentiable, is

Answer 1

$f\left(x\right)=|2x+1|-3|x+2|+|x^2+x-2|$

$f\left(x\right)=\{\begin{array}{cc}{x}^2-7;& x\ge 1\\ -x^2-2x-3;& -\frac{1}{2}\le x<1\\ -x^2-6x-5;& -2<x<-\frac{1}{2}\\ x^2+2x+3;& x\le -2\end{array}$

$\therefore f^{\prime }\left(x\right)=\{\begin{array}{cc}2x;& x>1\\ -2x-2;& -\frac{1}{2}<x<1\\ -2x-6;& -2<x<-\frac{1}{2}\\ 2x+2;& x<-2\end{array}$

Check at $1,-2$ and $\frac{-1}{2}$
Differentiable at $x=-2$ but not differentiable at $x=1$ and $\frac{-1}{2}.$