Question

Let f(x) = | |x| – 1|, then the point where f(x)is not differentiable, is / are?

• A. 0
• B. 1
• C. +1
• D. 0,±1

Answer 1

The Correct Option is D

Given the function:
\([ f(x) = | |x| - 1| ]\)

Let's break down the function step-by-step: 

1. For \(( |x| ):\) This function has a cusp (sharp turn) at ( x = 0 ). So, it's not differentiable at ( x = 0). 
2. For ( |x| - 1 ): This function is just a transformation of the absolute value function shifted downward by 1 unit. This will create two potential points of non-differentiability at ( x = 1 ) and ( x = -1 ) due to the absolute value function having a cusp at these points. 
3. For ( | |x| - 1| ): This outer absolute value further creates a cusp at any point where ( |x| - 1 = 0 ), which are the points ( x = 1 ) and ( x = -1 ).

Considering all the above observations, the function ( f(x) ) is not differentiable at ( x = 0, 1,) and ( -1 ). 
Hence, the correct option is D: 0,±1