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NDA 1 2024 Maths, question 84

Topic: Differential calculus

Passage · Q83 to Q84

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

Q84

Consider the following statements :

  1. f′′(e)=1ef''(e)=\frac{1}{e}

  2. f(x)f(x) attains local minimum value at x=ex=e

  3. A local minimum value of f(x)f(x) is ee

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. Differentiate again: f″(x) = (2 − ln x)/[x(ln x)³], so f″(e) = 1/e.
  2. Also f′ changes from negative to positive at e, establishing a local minimum, whose value is f(e) = e.
  3. All three statements are correct.