Let k be an integer. The number of integral values of k for which the equation 3x + 3 x = k has exactly 3 distinct solutions in the interval [0, ] is:

Step-by-step solution

Let f(x) = 3x + 3 x . We need to find the number of intersections of the graph of y = f(x) and the horizontal line y = k in the interval [0, ] . First, find the critical points by setting f'(x) = 0 : f'(x) = 3 3x + 3 x Using the identity 3x = 4 ^3 x - 3 x : f'(x) = 3(4 ^3 x - 3 x) + 3 x f'(x) = 12 ^3 x - 6 x f'(x) = 6 x(2 ^2 x - 1) = 6 x 2x Setting f'(x) = 0 in the interval [0, ] gives: x = 0 x = 2 2x = 0 2x = 2 , 3 2 x = 4 , 3 4 Now, evaluate f(x) at the endpoints and the critical points: f(0) = 0 f ( 4 ) = ( 3 4