Question

If \( f(x) = |x| - |1| \), then points where \( f(x) \) is not differentiable, is/are:

• A. \( 0, 1 \)
• B. \( \pm 1, 0 \)
• C. \( 0 \)
• D. \( 1 \text{ only} \)

Hint:

A function is not differentiable at points where sharp corners, vertical tangents, or discontinuities occur.

Answer 1

The Correct Option is B

Consider the function \( f(x) = ||x| - 1| \) and analyze the transformations step by step: First, examine \( y = |x| \): \[ \text{The graph of } y = |x| \text{ has a sharp corner at } x = 0. \] Next, for \( y = |x| - 1 \): \[ \text{The graph shifts vertically downward by 1 unit.} \] Finally, for \( f(x) = ||x| - 1| \): \[ \text{Sharp corners are formed at } x = \{-1, 0, 1\}. \] Thus, \( f(x) \) is not differentiable at the points \( \{-1, 0, 1\} \). Final Answer: \[ \boxed{\pm 1, 0} \]