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JEE MainMathematicsQuadratic Equation

The number of integral values of k for which the equation |x^2 - 4x + 3| - x = k has exactly three distinct real roots is

Options

  1. A1
  2. B0
  3. C2
  4. D3

Correct answer

A. 1

Step-by-step solution

Let f(x) = |x^2 - 4x + 3| - x . We can define f(x) piecewise by finding the roots of x^2 - 4x + 3 = 0 , which are x = 1 and x = 3 . f(x) = x^2 - 5x + 3 for x (- , 1] [3, ) f(x) = -x^2 + 3x - 3 for x (1, 3) Let us analyze the critical points and intervals of monotonicity for f(x) : For x (- , 1] , f'(x) = 2x - 5 For x (1, 3) , f'(x) = -2x + 3 . Setting f'(x) = 0 gives x = 1.5 . Here, f(x) increases from f(1) = -1 to a local maximum f(1.5) = -0.75 , and then decreases to f(3) = -3 . For x [3, ) , f'(x) = 2x - 5 > 0 ,

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