The number of points,
Question:

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

Solution:

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

$f(x)= \begin{cases}x^{2}-7 ; & x>1 \\ -x^{2}-2 x-3 ; & -\frac{1}{2}<x<1 \\ -x^{2}-6 x-5 ; & -2<x<\frac{-1}{2} \\ x^{2}+2 x+3 ; & x<-2\end{cases}$

$\therefore f^{\prime}(x)=\left\{\begin{array}{lc}2 x ; & x>1 \\ -2 x-3 ; & -\frac{1}{2}<x<1 \\ -2 x-6 ; & -2<x<\frac{-1}{2} \\ 2 x+2 ; & x<-2\end{array}\right.$

Check at $1,-2$ and $\frac{-1}{2}$

Non. Differentiable at $x=1$ and $\frac{-1}{2}$

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