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NDA 2 2025 Maths, question 99

Topic: Differential calculus

Q99

Consider the following statements regarding the function f(x)=1x−5f(x)=\frac{1}{x-5} :

Statement-I : f(x)f(x) is decreasing on the intervals x<5x<5 and x>5x>5.

Statement-II : f′(x)>0f'(x)>0 for all x≠5x\ne 5.

Which one of the following is correct in respect of the above statements?

  1. (a) Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
  2. (b) Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
  3. (c) Statement-I is correct but Statement-II is not correctCorrect answer
  4. (d) Statement-I is not correct but Statement-II is correct

Solution.

Answer: (c) Statement-I is correct but Statement-II is not correct

  1. f′(x)=−1/(x−5)²<0 wherever defined.
  2. Hence f decreases separately on each interval x<5 and x>5.
  3. Statement I is true and II is false; the discontinuity at 5 prevents treating the two intervals as one.