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Let f(x) = {(x - 1) sin(1/(x - 1)), x ≠ 1; 0, x = 1}. Then which one of the following is true?
- f is neither differentiable at x = 0 nor at x = 1
- f is differentiable at x = 0 and at x = 1
- f is differentiable at x = 0 but not at x = 1
- f is differentiable at x = 1 but not at x = 0
Correct answer: f is differentiable at x = 0 but not at x = 1
Solution
The function is continuous at x = 1, but its derivative does not exist there due to the oscillatory behavior of the sine term as x approaches 1. However, at x = 0, the function is smooth and differentiable.
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