Question

If $$ f(x) = 2 + |\sin^{-1} x|, $$ and $$ A = \{ x \in \mathbb{R} \mid f'(x) \text{ exists} \}, $$ then find $ A $.

• A. \( \{0\} \)
• B. \( [-1,1] \)
• C. \( (-\infty, -1) \cup (1, \infty) \)
• D. \( (-1, 0) \cup (0,1) \)

Hint:

Check points where inside function equals zero for differentiability of absolute value functions.

Answer 1

The Correct Option is D

Function involves absolute value of \( \sin^{-1} x \), which is not differentiable at points where \( \sin^{-1} x = 0 \), i.e., at \( x=0 \). Within \( (-1,1) \), \( \sin^{-1} x \) is differentiable. Hence, \( f'(x) \) exists everywhere except at \( x=0 \).