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MHT CET202526 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The function f (x)=[x(x-2)]^2 is increasing in the set

Options

  1. A(- , 0) (2, )
  2. B(- , 1)
  3. C(1,2)
  4. D(0,1) (2, )

Correct answer

D. (0,1) (2, )

Step-by-step solution

To determine where f(x)=[x(x-2)]^2 is increasing, we analyze the sign of its derivative. Rewriting gives f(x)=(x^2-2x)^2 . Differentiating yields f'(x)=2(x^2-2x)(2x-2)=4x(x-1)(x-2) . The critical points are x=0 , x=1 , and x=2 . Evaluating the sign of f'(x) in the intervals: On (- ,0) , f'(x) On (0,1) , f'(x)>0 (increasing). On (1,2) , f'(x) On (2, ) , f'(x)>0 (increasing). The function is increasing on (0,1) (2, ) , corresponding to option D.

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