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NDA 1 2024 Maths, question 83

Topic: Differential calculus

Passage · Q83 to Q84

Let f(x)=xln⁡x; (x>1)f(x)=\frac{x}{\ln x};\ (x>1)

Q83

Consider the following statements :

  1. f(x)f(x) is increasing in the interval (e,∞)(e,\infty )

  2. f(x)f(x) is decreasing in the interval (1,e)(1,e)

  3. 9ln⁡7>7ln⁡99\ln 7>7\ln 9

Which of the statements given above are correct ?

  1. (a) 1 and 2 only
  2. (b) 2 and 3 only
  3. (c) 1 and 3 only
  4. (d) 1, 2 and 3Correct answer

Solution.

Answer: (d) 1, 2 and 3

  1. For x > 1, f′(x) = (ln x − 1)/(ln x)².
  2. It is negative on (1, e) and positive on (e, ∞), proving statements 1 and 2.
  3. Since 9 > 7 > e, f(9) > f(7), or 9/ln 9 > 7/ln 7.
  4. Multiplication by positive logarithms gives 9 ln 7 > 7 ln 9.
  5. All three statements hold.