Find the coordinates of each turning point on the graph of \(y=3x^4-16x^3+18x^2\) and determine in each case whether it is a maximum point or a minimum point.

How can we determine if a turning point is a minimum or a maximum?

Sketch the graph of \(y=3x^4-16x^3+18x^2\), and state the set of values of \(k\) for which the equation \(3x^4-16x^3+18x^2=k\) has precisely two real roots for \(x\).

What does the line \(y=k\) look like on the graph?

We could try using a ruler as \(y=k\), and explore what happens when we add this to the graph of \(y=3x^4-16x^3+18x^2\) as we vary \(k\).