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 \(m=3b^2-1\).

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

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

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

  5. \(R\) 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 \(\dfrac{27}{4}\).