Let f(x) = _ k-4 (x^2 - kx + 2k) + 1 x^2 + 4x + k . If the domain of f(x) is the set of all real numbers R , then the sum of all possible integral values of k is

Step-by-step solution

For the domain of f(x) to be the set of all real numbers R , each term must be defined for all x R . For the logarithmic term _ k-4 (x^2 - kx + 2k) : 1. The base must be strictly positive and not equal to 1: k-4 > 0 k > 4 k-4 1 k 5 2. The argument must be strictly positive for all x R : x^2 - kx + 2k > 0 Since the coefficient of x^2 is positive, the discriminant must be strictly negative: D k^2 - 8k k(k-8) 0 For the rational term 1 x^2 + 4x + k : The expression inside the square root must be strictly positive for a