Consider the function f(x) = x^2 2 - 6x + 4 _e(x-1) defined for x > 1 , and the following statements: S₁ : f(x) has a local maximum at x=2 and a local minimum at x=5 . S₂ : The graph of f(x) changes its concavity at x=3 . S₃ : The equation f(x) = -8 has exactly two distinct real roots. S₄ : The equation f(x) = -11 has exactly three distinct real roots. (Given _e 2 0.693 ) Which of the following options is correct?

Step-by-step solution

Given f(x) = x^2 2 - 6x + 4 _e(x-1) for x > 1 First derivative: f'(x) = x - 6 + 4 x-1 = x^2 - 7x + 10 x-1 = (x-2)(x-5) x-1 Critical points are x=2 and x=5 . For x (1, 2) , f'(x) > 0 (increasing). For x (2, 5) , f'(x) For x (5, ) , f'(x) > 0 (increasing). Therefore, f(x) has a local maximum at x=2 and a local minimum at x=5 . Statement S₁ is correct. Second derivative: f''(x) = 1 - 4 (x-1)^2 = (x-1)^2 - 4 (x-1)^2 = x^2 - 2x - 3 (x-1)^2 = (x-3)(x+1) (x-1)^2 For x > 1 , f''(x) = 0 at x=3 . f''(x) 0 for x (3, ) . The c