2024 HCI P1 Q7

It is given that \(\mathrm{f}\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx\), where \(a\), \(b\) and \(c\) are non-zero real constants.

1. Show that \(\mathrm{f}\left( -x \right)=-\mathrm{f}\left( x \right)\).[1]
2. It is given that \(\mathrm{f}\left( x \right)=0\) has only one real root and one of the non-real roots is \(p+q\mathbf{i}\), where \(p\) and \(q\) are non-zero real constants. Find, in terms of \(p\) and \(q\), all the other non-real roots of \(\mathrm{f}\left( x \right)=0\), justifying your answers.[3]
3. Given that \(\mathop{\displaystyle\displaystyle \displaystyle\int }_{0}^{3}\mathrm{f}\left( x \right)\mathrm{d}x=-5\), state the values of \(\displaystyle\displaystyle \displaystyle\int_{-3}^{3}{\mathrm{f}\left( x \right)\mathrm{d}x}\) and \(\displaystyle\displaystyle \displaystyle\int_{-3}^{3}{\mathrm{f}\left( \left| x \right| \right)\mathrm{d}x}\).[2]

Let \(a=1\) and \(b=3\) .

4. By considering \(f\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)\), find the range of values of \(c\) such that the curve with equation \(y=\mathrm{f}\left( x \right)\) has 2 stationary points, showing your working clearly.[3]