9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa025dee-b19f-4743-b212-2fff9a868eaf-18_689_830_127_646}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a quartic expression in \(x\).
The curve
- has maximum turning points at \(( - 1,0 )\) and \(( 5,0 )\)
- crosses the \(y\)-axis at \(( 0 , - 75 )\)
- has a minimum turning point at \(x = 2\)
- Find the set of values of \(x\) for which
$$\mathrm { f } ^ { \prime } ( x ) \geqslant 0$$
writing your answer in set notation.
Find the equation of \(C\). You may leave your answer in factorised form.
The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant.
Given that the graph of \(C _ { 1 }\) intersects the \(x\)-axis at exactly four places,find the range of possible values for \(k\).
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