Let S be the set of all a R for which the angle between the vectors u = a (2^x + 2^ -x ) i - a j + 2 k and v = (2^x + 2^ -x ) i + 2(2^x + 2^ -x ) j - 4 k is obtuse for all real x . Then S is equal to

Step-by-step solution

For the angle between u and v to be obtuse, their dot product must be strictly negative. u v a(2^x + 2^ -x )^2 - 2a(2^x + 2^ -x ) - 8 Let t = 2^x + 2^ -x . By AM-GM inequality, t 2 for all real x . The condition becomes f(t) = at^2 - 2at - 8 Case 1: a > 0 The parabola opens upwards. As t , f(t) . Thus, f(t) Case 2: a = 0 f(t) = -8 Case 3: a The parabola opens downwards. The vertex is at t = -(-2a) 2a = 1 . For t 2 , the function f(t) is strictly decreasing. Therefore, the maximum value of f(t) on [2, ) occurs at t