Express \(\dfrac{3x}{(x-1)(x+2)}\) in partial fractions.

Show that \(\dfrac{dy}{dx}\) is negative at all points on the graph of \[y=\frac{3x}{(x-1)(x+2)}.\]

How does using partial fractions help us to differentiate the expression?

Sketch this graph, showing the two asymptotes parallel to the \(y\)-axis and the asymptote perpendicular to the \(y\)-axis.

Things to look out for when curve sketching:

  • asymptotes;
  • intercepts;
  • stationary points and their nature;
  • behaviour as \(x\rightarrow \pm \infty\).

By sketching on the same diagram a second graph (the equation of which should be stated), or otherwise, find the number of real roots of the equation \[(x-1)(x+2)(x+3)=3x.\]

Can we make this look something like the graph we’ve just sketched?

Could we make the roots of the equation into the intersections between the original graph and the new graph?