I am given an equation $y = x(x-2)^2$ and I am asked to find the set of values of $k$ such that $y=k$ has 3 real roots.
I already differentiated the equation and got $3x^2-8x+4$. And I also calculated the minimum and maximum values. When $x=2$, it is a minimum and when $x=2/3$, it's a maximum.
I think you've got almost everything you need. Next, find the $y$ values at the local minimum and local maximum.
Your polynomial increases until it reaches the local maximum, decreases to the local minimum and increases again after that. For any value of $k$, $y=k$ can have at most one root in each of these sections, and the only way to have roots in all three is if $k$ is between the minimum and the maximum.