Prove, by considering the turning points on the graph \(y=x^3-3a^2x+b\) or otherwise, that if the equation \(x^3-3a^2x+b=0\) has three distinct real roots, then \(4a^6>b^2\).
Where are the turning points of \(y=x^3-3a^2x+b\), and how many are there?
What is the effect of \(a\) being negative rather than positive? What happens when \(a=0\)?
What happens to the number of roots as we translate the curve in the \(y\)-direction?