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NDA2025MathematicsApplication of DerivativesActual

For the following two (02) items: Consider the function f(x) = 1 - [3] (x-1)^2 . The function has

Options

  1. Aa minimum at x = 1
  2. Ba maximum at x = 1
  3. Cneither maximum nor minimum at x = 1
  4. Dno extremum

Correct answer

B. a maximum at x = 1

Step-by-step solution

Given f(x) = 1 - (x-1)^ 2/3 Differentiating with respect to x : f'(x) = - 2 3 (x-1)^ -1/3 = -2 3(x-1)^ 1/3 For x 0 . Thus, f(x) is strictly increasing. For x > 1 , x-1 > 0 f'(x) Since f'(x) changes sign from positive to negative as x passes through 1 , f(x) has a local maximum at x = 1 . Alternatively, (x-1)^ 2/3 0 for all real x . 1 - (x-1)^ 2/3 1 f(x) f(1) for all x . Hence, f(x) has a maximum at x = 1 . Answer: a maximum at x = 1

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