Question

Let $f:R\to R$ be defined as $f\left(x\right)=|x|+|x^2-1|.$ The total number of points at which $f$ attains either a local maximum or a local minimum is

Answer 1

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

Case $\left(i\right)$ If $x\ge 1,$ then
$f\left(x\right)=x+(x-1)(x+1)\Rightarrow f\left(x\right)=x^2+x-1$

Case $\left(ii\right)$ If $0\le x<1,$ then
$f\left(x\right)=x+(1-x)(x+1)\Rightarrow f\left(x\right)=1-x^2+x$

Case $\left(iii\right)$ If $-1<x<0,$ then
$f\left(x\right)=-x+(1-x)(x+1)\Rightarrow f\left(x\right)=1-x^2-x$

Case $\left(iv\right)$ If $x\le -1,$ then
$f\left(x\right)=-x-(1-x)(x+1)\Rightarrow f\left(x\right)=x^2-x-1$

Plotting the graph for this function


we can see $3$ points of local minima and $2$ points of local maxima