Question

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

Answer 1

Answer: 5

$f^{\prime }(x)=\frac{|x|}{x}+\frac{|x^2-1|}{x^2-1}\cdot (2x)$
$=<br/><br/>\{\begin{array}{ll}<br/><br/>2x-1,& x<-1\\ <br/><br/>-(2x+1),& -1<x<0\\ <br/><br/>1-2x,& 0<x<1\\ <br/><br/>2x+1& x>1<br/><br/>\end{array}$

So, $f^{\prime }(x)$ changes sign at points
$x=-1,-\frac{1}{2},0,\frac{1}{2},1$
so, total number of points of local maximum or minimum is $5.$