The function has
For the following two (02) items: Consider the function f(x)=1-\sqrt{(x-1)^{2}}.
- A. a minimum at x=1
- B. a maximum at x=1 ✓
- C. neither maximum nor minimum at x=1
- D. no extremum
Correct Answer: B. a maximum at x=1
Explanation
Compute the derivative: f'(x) = -\frac{2}{3}(x-1)^{-1/3}. At x=1, f'(x) is undefined, making it a critical point. For x \lt 1, (x-1)^{-1/3} is negative, so f'(x) \gt 0 (increasing). For x \gt 1, (x-1)^{-1/3} is positive, so f'(x) \lt 0 (decreasing). Since the function changes from increasing to decreasing, it has a local maximum at x=1.
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