Consider the following statements regarding the function f(x)=\frac{1}{x-5}: Statement-I: f(x) is <strong>DECREASING</strong> on the intervals x \lt 5 and x \gt 5. Statement-II: f'(x) \gt 0 for <strong>ALL</strong> x \neq 5. Which one of the following is correct in respect of the above statements?
- A. Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
- B. Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
- C. Statement-I is correct but Statement-II is not correct ✓
- D. Statement-I is not correct but Statement-II is correct
Correct Answer: C. Statement-I is correct but Statement-II is not correct
Explanation
The derivative is f'(x) = -\frac{1}{(x-5)^2}. For all x \neq 5, the term (x-5)^2 \gt 0, so f'(x) \lt 0. Since f'(x) \lt 0, the function is strictly decreasing on (-\infty, 5) and (5, \infty). Therefore, Statement-I is correct but Statement-II is incorrect because it incorrectly claims f'(x) \gt 0.
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