The curve \(y=x^3-6x^2+kx+1\) is defined for all real \(x\), where \(k\) is a constant.
Find the values of \(k\) for which the curve has two distinct stationary points, exactly one stationary point, and no stationary points.[3]
For the case with exactly one stationary point, find its coordinates and determine its nature.[3]
Find all values of \(k\) for which the function is strictly increasing for all real \(x\). Explain why the boundary value is included.[3]