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NDA 2 2025 Maths, question 82

Topic: Differential calculus

Passage · Q81 to Q82

Consider the function f(x)=1−(x−1)23f(x)=1-\sqrt[3]{(x-1)^{2}}.

Q82

The function has

  1. (a) a minimum at x=1x=1
  2. (b) a maximum at x=1x=1Correct answer
  3. (c) neither maximum nor minimum at x=1x=1
  4. (d) no extremum

Solution.

Answer: (b) a maximum at x=1x=1

  1. Since the real cube root of (x−1)² is nonnegative, f(x)=1−|x−1|^(2/3)≤1, with equality only at x=1.
  2. Hence x=1 is a maximum despite the derivative not existing there.