Let the function f(x) = x^2 + ax + b x - c , x c , be strictly increasing in (- , -1) (3, ) and strictly decreasing in (-1, 1) (1, 3) . If the local maximum value of f(x) is -2 , then the value of f(2) is :

Step-by-step solution

Given f(x) = x^2 + ax + b x - c The interval of decrease is split at x = 1 , which indicates a vertical asymptote at x = 1 . Thus, c = 1 . So, f(x) = x^2 + ax + b x - 1 Differentiating with respect to x : f^ (x) = (2x + a)(x - 1) - (x^2 + ax + b)(1) (x - 1)^2 f^ (x) = x^2 - 2x - (a + b) (x - 1)^2 The critical points are the boundaries of the intervals of monotonicity, which are x = -1 and x = 3 . These must be the roots of the numerator x^2 - 2x - (a + b) = 0 . Product of roots: (-1)(3) = -(a + b) a + b = 3 By the