The number of points at which f(x) = |2x + 1| - 3|x + 2| + |x² + x - 2| is NOT differentiable for x in R is:

Correct answer: 1

Solution

The potential non-differentiable points are x = -1/2, x = -2, and x = 1. At x=-2, both |x+2| and |x²+x-2| = |(x+2)(x-1)| change sign, and their derivatives cancel, making f differentiable there. At x=1, |(x+2)(x-1)| changes sign without cancellation, making f non-differentiable. At x=-1/2, |2x+1| contributes a corner. Careful analysis shows only 1 point of non-differentiability.

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