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NDA 1 2024 Maths, question 91

Topic: Differential calculus

Passage · Q91 to Q92

A differentiable function f(x)f(x) has a local maximum at x=0x=0. Let y=2f(x)+ax−by=2f(x)+ax-b.

Q91

Which of the following is/are correct ?

  1. f′(0)=0f'(0)=0

  2. f′′(0)<0f''(0)<0

Select the correct answer using the code given below :

  1. (a) 1 onlyCorrect answer
  2. (b) 2 only
  3. (c) Both 1 and 2
  4. (d) Neither 1 nor 2

Solution.

Answer: (a) 1 only

  1. Fermat’s theorem gives f′(0) = 0 at an interior local maximum of a differentiable function.
  2. A negative second derivative is sufficient but not necessary: f(x) = −x⁴ has a local maximum at 0 and f″(0) = 0.
  3. Thus only statement 1 is guaranteed.