Let p = ( + 3) i - 8 j + ( - 3) k and q = i + j + k be two vectors, where [0, 2 ] and is a real parameter. If the angle between p and q is obtuse for all possible values of , then the maximum integer value of is

Step-by-step solution

For the angle between p and q to be obtuse, their dot product must be strictly negative: p q ( + 3) ^2 - 8 + ( - 3) Let x = . Since [0, 2 ] , x [-1, 1] . Let f(x) = ( + 3)x^2 - 8x + ( - 3) . We require f(x) For f(x) to be negative on [-1, 1] , its values at the boundaries must be negative: f(1) = ( + 3) - 8 + ( - 3) = 2 - 8 f(-1) = ( + 3) + 8 + ( - 3) = 2 + 8 Thus, we must have When The vertex of the parabola is at x_v = -(-8) 2( + 3) = 4 + 3 . For If x_v [-1, 1] (i.e., x_v If x_v [-1, 1] , the maximum value is f(x