The graphs of \(y = x^3 - x\) and \(y = m(x-a)\) are drawn on the axes below. Here \(m>0\) and \(a \le -1\).

The line \(y=m(x-a)\) meets the \(x\)-axis at \(A = (a,0)\), touches the cubic \(y=x^3 -x\) at \(B\) and intersects again with the cubic at \(C\). The \(x\)-coordinates of \(B\) and \(C\) are respectively \(b\) and \(c\).

Graph of the cubic and the line. The line touches the cubic before its first turning point then goes on to cross it after the cubic's second turning point.
  1. Use the fact that the line and cubic touch when \(x=b\), to show that \(m=3b^2-1\).

What do we know about curves and lines which touch at some point? How can that help us to answer this question?

  1. Show further that \[a = \frac{2b^3}{3b^2 - 1}.\]

Can we find \(m\) simply in terms of \(a\) and \(b\)? Could we drop a vertical from \(B\)?

  1. If \(a=-10^6\), what is the approximate value of \(b\)?

Can you sketch a graph to show this situation?

  1. Using the fact that \[x^3 - x - m(x-a) = (x-b)^2 (x-c)\] (which you need not prove), show that \(c=-2b\).

Why is this true? For which values of x is this true? So this tells us that …

  1. \(R\) is the finite region bounded above by the line \(y=m(x-a)\) and bounded below by the cubic \(y = x^3 - x\). For what value of \(a\) is the area of \(R\) largest?

Show that the largest possible area of \(R\) is \(\dfrac{27}{4}\).

How does the area of \(R\) change as \(a\) changes?

Or… can we find a formula for the area? When we’ve done so, how can we maximise this?