Let f(x) = 2x + x - k , where k is a real parameter. If the equation f(x) = 0 has exactly one real root in the interval [0, ] , then the complete set of values of k is

Step-by-step solution

f(x) = 2x + x - k Differentiating with respect to x , we get: f^ (x) = 2 - x Since -1 x 1 for all x , we have f^ (x) 1 > 0 . Thus, f(x) is a strictly increasing function on the interval [0, ] . For f(x) = 0 to have exactly one real root in [0, ] , the function must change sign (or touch zero) at the endpoints of the interval. Therefore, by the Intermediate Value Theorem: f(0) f( ) 0 Evaluating the function at the endpoints: f(0) = 2(0) + (0) - k = 1 - k f( ) = 2 + ( ) - k = 2 - 1 - k Substituting these into the con