I have no idea what you mean by "think k= 6". Why should we think that? If I were asked to find a value of k such that f is a pdf, I would use the fact that the integral 0 to 1 gives the total probability that x is between 0 and 1 which is clealy 1.0. What do you get when you integrate x(1- x) from 0 to 1?

Your next two questions have missing information. You are essentially asked for therangeof the two function, but that depends on the domain, the possible values of x, but I see no domain given here. Normally the "default domain" would be "all values of x for which the functioncanbe calculated" which would be "all real numbers" for (a) and "all real numbers except 0" for (b) but because this is a probability problem, that cannot be true. The integral of each of these functions over the default domain is NOT 1.