To determine the correctness of the given statements about the function \(f(x) = \frac{1}{x-5}\), we need to analyze the behavior of this function.

Step-by-Step Analysis:

• Critical Point and Domain:
  • The function \(f(x) = \frac{1}{x-5}\) is undefined when \(x = 5\) because the denominator becomes zero.
  • The domain of the function is \(x \neq 5\).
• Statement-I Analysis:
  • The function \(f(x)\) can be analyzed separately for the intervals \(x < 5\) and \(x > 5\).
  • To check for monotonicity (increasing/decreasing behavior), compute the derivative \(f'(x)\):

\(f'(x) = \frac{-1}{(x-5)^2}\)

•  
  • The derivative \(f'(x) = \frac{-1}{(x-5)^2}\) is always negative for all \(x \neq 5\) since the square of any real number is positive, and dividing -1 by a positive number results in a negative value.
  • This implies \(f(x)\) is decreasing on both intervals: \(( -\infty, 5 )\) and \(( 5, \infty )\).
  • Thus, Statement-I is correct.
• Statement-II Analysis:
  • Statement-II claims \(f'(x) > 0\) for all \(x \neq 5\). However, from our computation, \(f'(x) = \frac{-1}{(x-5)^2}\) is negative for all \(x \neq 5\).
  • This invalidates Statement-II because the derivative is not positive.

Conclusion:

• Statement-I is correct as \(f(x)\) is decreasing on both specified intervals.
• Statement-II is incorrect since \(f'(x)\) is negative for all defined \(x\).

Thus, the correct answer is: Statement-I is correct but Statement-II is not correct.