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Let f(x) = |1 - x|. Define g(x) = arcsin(f(|x|)). Find the number of points where g(x) is non-differentiable.

1. 2
2. 3
3. 4
4. 5

Correct answer: 3

Solution

g(x) = arcsin(|1-|x||). Domain: need |1-|x|| in [-1,1], so 0 <= |1-|x|| <= 1 (since absolute value is non-negative). This means 0 <= 1-|x| <= 1 or -1 <= 1-|x| <= 0, giving 0 <= |x| <= 1 or 1 <= |x| <= 2. So domain is [-2,-1] union [-1,1] union [1,2] = [-2,2]. Potential non-differentiable points: x=0 (corner of |x|), x=+/-1 (corners of |1-|x||), x=+/-2 (endpoints where arcsin argument = 1 and derivative is infinite, but these are domain boundaries). Since the domain is closed interval [-2,2], end points x=+/-2 are included but derivatives from both sides don't exist — typically we check interior points: x = -1, 0, 1. At these three points g is non-differentiable.

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