Let \(f(x) = x^3 + ax^2 + bx + c\), where the coefficients \(a\), \(b\) and \(c\) are real numbers. The figure below shows a section of the graph of \(y=f(x)\). The curve has two distinct turning points; these are located at \(A\) and \(B\), as shown. (Note that the axes have been omitted deliberately.)

- Find a condition on the coefficients \(a\), \(b\), \(c\) such that the curve has two distinct turning points if, and only if, this condition is satisfied.

*It may be assumed from now on that the condition on the coefficients in (i) is satisfied.*

Let \(x_1\) and \(x_2\) denote the \(x\) coordinates of \(A\) and \(B\), respectively. Show that \[x_2 - x_1 = \frac{2}{3} \sqrt{a^2 - 3b}.\]

Suppose now that the graph of \(y=f(x)\) is translated so that the turning point at \(A\) now lies at the origin. Let \(g(x)\) be the cubic function such that \(y=g(x)\) has the translated graph. Show that \[g(x) = x^2 \left(x - \sqrt{a^2 - 3b}\right).\]

Let \(R\) be the area of the region enclosed by the \(x\)-axis and the graph \(y=g(x)\). Show that if \(a\) and \(b\) are rational then \(R\) is also rational.

Is it possible for \(R\) to be a non-zero rational number when \(a\) and \(b\) are both irrational? Justify your answer.