The number of integer values of c for which the equation x^3 - 3x^2 - 9x + c = 0 has exactly 3 distinct real roots is

Step-by-step solution

Let f(x) = x^3 - 3x^2 - 9x + c . For the cubic equation f(x) = 0 to have exactly 3 distinct real roots, the function must have a local maximum that is strictly positive and a local minimum that is strictly negative. First, find the critical points by setting f^ (x) = 0 : f^ (x) = 3x^2 - 6x - 9 = 0 3(x^2 - 2x - 3) = 0 3(x - 3)(x + 1) = 0 The critical points are x = -1 and x = 3 . By the second derivative test or sign change of f^ (x) , x = -1 is a point of local maximum and x = 3 is a point of local minimum. Evaluat