The diagram shows the graph of the function \(y=\mathrm{f}(x)\) where
\[\mathrm{f}(x)=\frac{ax+b}{x^2+cx+d}\]
with \(\mathrm{f}\left(-\dfrac52\right)=0,\ \mathrm{f}(0)=\dfrac54\). The lines \(x=-4,\ x=-1\) and the \(x\)-axis are all asymptotes.
Using the information given, find the value of \(c\) and of \(d\) and show that \(a=2,\ b=5\).[4]
Using the values of \(a,\ b,\ c,\) and \(d\) found in (a), find the value of \(A\) and of \(B\) for which \(\mathrm{f}(x)=\dfrac A{x+4}+\dfrac B{x+1}\).[3]
Show that \(\left(-\dfrac52,0\right)\) is a point of inflexion.[3]
State the set of values of \(x\) for which \(\mathrm{f}'(x)<0\).[2]
State the set of values of \(x\) for which \(\mathrm{f}''(x)>0\).[2]
Sketch the graph of \(y=\mathrm{f}'(x)\), clearly indicating any intercepts with the axes, the coordinates of any local maximum or minimum points and the equations of the asymptotes (if any).[4]