Let \(\mathrm f(x)=b(x+1)\mathrm e^{-2x}\), where \(b>0\), \(x\in\mathbb R\).
Find \(\mathrm f'(x)\) in terms of \(b\).
Hence find the coordinates of the maximum point of \(y=\mathrm f(x)\) in terms of \(b\).[4]
Given that \(\mathrm f(0)=3\), show that \(b=3\).
Sketch the graph of \(y=\mathrm f(x)\), indicating clearly the coordinates of the turning points, axial intercepts and the equation of the asymptote.
State \(\displaystyle\lim_{x\to\infty}\mathrm f(x)\).
Find the coordinates of the point of inflexion and justify your answer.[9]
Using \(b=3\), find the exact range of values of \(k\) for which \([\mathrm f(x)]^2=k\) has three distinct real roots.[3]