FM 395 P1 Q1

The curve \(C\) has equation \(y=\frac{{{x}^{2}}+3x+b}{x+a}\) \((x\ne -a)\), where \(a\) and \(b\) are constants. It is given that the line \(y=x+4\)is an asymptote of \(C\), and that \(C\) has a stationary point at \(x=2\). Show that \(a=-1\), and find the value of \(b\).[6]

Using the same values of \(a\) and \(b\),

1. Show that \(C\) has exactly two stationary points, and find the coordinates of each stationary point,[3]
2. Draw a sketch of \(C\),[2]

Deduce that, for all positive values of \(k\), the equation \({{x}^{3}}+3{{x}^{2}}+(b-k)x-ka=0\) has exactly one negative root.