The answer is all k in the closed interval [0,1]. Informally, the reasoning is as follows. For k > 1, and large values of x the derivative is quite large and so a small change in x values produces large change in the function values. Similarly for k < 0 and x values close to 0. For k between 0 and 1, the function is well behaved on [0,1] and its graph gets quite flat for large values of x.
Here is "2/3" of the actual proof: