Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of \(x ^ { 3 }\)
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The function \(\mathrm { f } ( x )\) is defined by
$$\mathrm { f } ( x ) = x ^ { 3 } + 3 p x ^ { 2 } + q$$
where \(p\) and \(q\) are constants and \(p > 0\)
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Show that there is a turning point where the curve crosses the \(y\)-axis.
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(ii) The equation \(\mathrm { f } ( x ) = 0\) has three distinct real roots.
By considering the positions of the turning points find, in terms of \(p\), the range of possible values of \(q\).