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JEE MainMathematicsApplication of Derivatives

Consider the function f(x) = |x^2 - 1| - 2|x - 1| defined on R . Three statements are given below: (I) f is differentiable at x = 1 . (II) f is non-differentiable at exactly one point. (III) f has a local maximum at x = -1 . Which of the following options is correct?

Options

  1. AAll (I), (II) and (III) are TRUE.
  2. BOnly (I) and (III) are TRUE.
  3. COnly (II) and (III) are TRUE.
  4. DOnly (I) and (II) are TRUE.

Correct answer

D. Only (I) and (II) are TRUE.

Step-by-step solution

The critical points where the arguments of the moduli become zero are x = 1 and x = -1 . The domain is divided into three intervals: (- , -1) , [-1, 1) , and [1, ) . For x 1 , x^2 - 1 0 and x - 1 0 . f(x) = (x^2 - 1) - 2(x - 1) = x^2 - 2x + 1 = (x - 1)^2 . f'(x) = 2(x - 1) . The right-hand derivative at x = 1 is f'(1^+) = 0 . For -1 x f(x) = -(x^2 - 1) - 2(-(x - 1)) = -x^2 + 1 + 2x - 2 = -x^2 + 2x - 1 = -(x - 1)^2 . f'(x) = -2(x - 1) . The left-hand derivative at x = 1 is f'(1^-) = 0 . The right-hand derivative at

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