If the maximum value of the function f(x) = 2 x + k 2x on the interval [0, 2 ] is 9 4 , then the value of k is:

Step-by-step solution

Given f(x) = 2 x + k 2x . Differentiating with respect to x : f'(x) = 2 x - 2k 2x f'(x) = 2 x - 4k x x = 2 x(1 - 2k x) For critical points, f'(x) = 0 : x = 0 x = 2 (since x [0, 2 ] ) or 1 - 2k x = 0 x = 1 2k Let us evaluate f(x) at the critical points and the boundaries x = 0 and x = 2 . At x = 0 : f(0) = 2(0) + k(1) = k At x = 2 : f ( 2 ) = 2(1) + k(-1) = 2 - k If the critical point x = 1 2k lies in (0, 2 ) , then 2x = 1 - 2 ^2 x = 1 - 1 2k^2 . The value of the function at this point is: f(x) = 2 ( 1 2k ) + k (1 -