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MHT CET202523 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

The maximum value of x^ 2 / 3 +(x-2)^ 2 / 3 is

Options

  1. A0
  2. B2
  3. C2^ 2 3
  4. D1

Correct answer

B. 2

Step-by-step solution

Given function: f(x) = x^ 2/3 + (x-2)^ 2/3 The derivative is computed using the power rule: f'(x) = 2 3 x^ -1/3 + 2 3 (x-2)^ -1/3 = 2 3 ( 1 x^ 1/3 + 1 (x-2)^ 1/3 ) Critical points occur where f'(x) is undefined or zero. The derivative is undefined at x = 0 and x = 2 , which are critical points. Setting f'(x) = 0 yields 1 x^ 1/3 = - 1 (x-2)^ 1/3 . Cubing both sides gives x-2 = -x , so x = 1 . Evaluating at these points: f(0) = [3] 4 1.587 f(1) = 2 f(2) = [3] 4 1.587 The derivative sign analysis reveals: For x For 0

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