2018 EJC P1 Q7

It is given that \(\mathrm{f}\left( x \right)=2{{x}^{6}}-4{{x}^{4}}-6{{x}^{2}}-7.\) The diagram shows the curve with equation \(y=\mathrm{f}\left( x \right)\) for \(x\ge 0.\) The curve crosses the positive \(x\)-axis at \(x=\alpha\).

1. Find the value of \(\alpha ,\) giving the answer correct to \(3\) decimal places.[1]
2. Show that \(\mathrm{f}\left( x \right)=\mathrm{f}\left( -x \right)\) for all real values of \(x.\)What can be said about the six roots of the equation \(\mathrm{f}\left( x \right)=0?\)[4]

It is given that \(\mathrm{{g}'}\left( x \right)=\mathrm{f}\left( x \right),\) for all real values of \(x.\)

3. Determine the \(x\)-coordinates of all the stationary points of graph of \(y=\mathrm{g}\left( x \right)\)and determine their nature.[3]
4. For which values of \(x\) is the graph of \(y=\mathrm{g}\left( x \right)\) concave upwards?[3]