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Let f(x) = { -1, -2 ≤ x < 0; x² - 1, 0 ≤ x ≤ 2 } and g(x) = |f(x)| + f(|x|). Then, in the interval (-2, 2), g is: (1) non continuous (2) differentiable at all points (3) not differentiable at two points (4) not differentiable at one point
- non continuous
- differentiable at all points
- not differentiable at two points
- not differentiable at one point
Correct answer: not differentiable at one point
Solution
The function g(x) is composed of f(x) and its absolute value, which introduces a potential point of non-differentiability. Specifically, g(x) is not differentiable at x = 0, where f(x) transitions from a constant to a quadratic function, creating a cusp.
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