let f: $R\rightarrow R $ defined by f(x) = min?

Question

let f: $R\rightarrow R $ defined by $f(x) = min (|x|,x^2-1)$,then f is

which one is correct ?

a) Differntiable everywhere except at one point

b) differntiable everywhere except at two points

i think option a ) is correct because f(x) = min$(|x|,x^2 -1)$= $|x|$ which is not diferentiable at 0

is it correct ???

Answer 1

You could try graphing both functions $|x|$ and $x^2 -1 $ on the same axes. Then you will be able to see what $f(x)$ looks like, and identify any suspicious points.

Answer 2

Hint:

note that the equation $|x|=x^2-1$ has solutions $x=\pm \frac{\sqrt{5}+1}{2}$. So these are the ''critical points'' where the minimum change from a function to the other.