Let `I RvecI R` be defined as `f(x)=|x|++x^2-1|dot` The total number of points at which `f` attains either a local maximum or a local minimum is_______

Options

A. 2
B. 4
C. 5
D. 6

Answer

C

Explanation

We have
`f(x)=|x|+|x^2-1|={{:(-x+x^2-1","x le -1),(-x-x^2+1","-1 x lt 0),(x-x^2+1"," 0le x lt 1),(x+x^2-1","x ge 1):}`
`rArr f(x)={:{(2x-1"," x lt -1),(-2x -1","-1 lt x lt 0),(-2x+1 "," 0 lt x lt 1),(2x +1 "," x lt 1):}`
Clearly f(x) is not differetaible at x = -1 ,0,1
The chages in sings of f(x) for different values of x are shown is Fig. 31

So f(x) changes its sign at 5 points
Hence , total number of points of local maximum of local minimum of f(x)is 5.