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

let f: IR → IR be defined as f(x) = |x| + |x² - 1|. The total number of points at which f attains either a local maximum or a local minimum is

$f(x)=|x|+|x^2-1|$

$\text{To find: total number of local maxima/minima of } f$

$|x|=0 \Rightarrow x=0;\quad x^2-1=0 \Rightarrow x=\pm 1$

$\text{breakpoints: } -1,\ 0,\ 1$

f(x)= x²-x-1, x<-1 -x²-x+1, -1≤ x<0 -x²+x+1, 0≤ x<1 x²+x-1, x≥ 1

$\text{Plan: } f' \text{ on each piece; check sign change at } -1,0,1$

$x<-1:\ f'=2x-1;\qquad -1<x<0:\ f'=-2x-1$

$0<x<1:\ f'=-2x+1;\qquad x>1:\ f'=2x+1$

$x<-1:\ f'=2x-1<0;\qquad x>1:\ f'=2x+1>0$

$-1<x<0:\ f'=-2x-1=0 \Rightarrow x=-\dfrac{1}{2},\quad f''<0 \Rightarrow \text{local max}$

$0<x<1:\ f'=-2x+1=0 \Rightarrow x=\dfrac{1}{2},\quad f''<0 \Rightarrow \text{local max}$

$x=-1:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$x=0:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$x=1:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$\text{maxima}=2,\qquad \text{minima}=3 \;\Rightarrow\; \text{total}=5$

$\text{Total number of local extrema} = 5$