2021 A-Level H2 Mathematics

A function \(\mathrm{f}\) is defined by \(\mathrm{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d\). The graph of \(y=\mathrm{f}\left( x \right)\) passes through the points \(\left( 1,5 \right)\) and \(\left( -1,-3 \right)\). The graph has a turning point at \(x=1\), and \(\displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)}\mathrm{d}x=6\).

Find the values of \(a\), \(b\), \(c\) and \(d\).[5]