14 The graph of \(y = x ^ { 3 } - 3 x\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-22_718_771_370_632}
The two stationary points have \(x\)-coordinates of - 1 and 1
The cubic equation
$$x ^ { 3 } - 3 x + p = 0$$
where \(p\) is a real constant, has the roots \(\alpha , \beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
14
- Explain why \(\alpha + \beta = - \gamma\)
14 - Find the set of possible values for the real constant \(p\).
14 - \(\quad \mathrm { f } ( x ) = 0\) is a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\)
14 - Show that the constant term of \(\mathrm { f } ( x )\) is \(p + 2\)
14
- (ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm { f } ( x )\)
\includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-24_2488_1719_219_150}
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Additional page, if required.
Write the question numbers in the left-hand margin.