The curve \(y=\left(\dfrac{x-a}{x-b}\right)e^x,\) where \(a\) and \(b\) are constants, has two stationary points. Show that \[a-b<0 \quad \text{ or } \quad a-b>4.\]

Show that, in the case \(a=0\) and \(b=\dfrac{1}{2}\), there is one stationary point on either side of the curve’s vertical asymptote, and sketch the curve.

Sketch the curve in the case \(a=\dfrac{9}{2}\) and \(b=0\).