The graph of \(y = f ( x )\), defined for \(- 3 \leq x \leq 7\), is shown below, along with the coordinates of the turning points and endpoints:
\includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-02_1157_1584_1539_319}
How many solutions are there to \(f ( x ) = 1\) ?
If \(f ( x ) = k\) has three distinct solutions, find the possible values of \(k\).
How many solutions are there to \(f \left( x ^ { 2 } \right) = 1\) ?
If \(f \left( x ^ { 2 } \right) = k\) has five distinct solutions, find the value of \(k\).
How many solutions are there to \([ f ( x ) ] ^ { 2 } = 2\) ?
If \([ f ( x ) ] ^ { 2 } = k\) has six distinct solutions, find the range of possible values of \(k\).
How many solutions are there to \(\log _ { 2 } f ( x ) = - 2025\) ?
How many solutions are there to \(\log _ { 2 } \left( [ f ( x ) ] ^ { 2 } \right) = 0\) ?
Show that, if \(n\) is a non-negative integer, \(4 ^ { 3 n } + 5 ^ { 2 n + 2 }\) cannot be a prime. [0pt]
[6] [0pt]
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All of these questions concern the curve \(y = g ( x )\).
Part of the graph of \(y = g ^ { \prime \prime } ( x )\) is shown below:
\includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-06_1083_1744_258_258}
You are given that \(y = g ( x )\) has exactly two local minima and one local maximum in this range.
Identify which of the labelled points could correspond to the local maximum.
Identify two of the labelled points which could correspond to the local minima.
There is more than one possible pair of answers but you are only required to give one.
Identify all of the labelled points which correspond to points of inflection.
As \(x \rightarrow - \infty , g ^ { \prime \prime } ( x ) \rightarrow 0\).
What does this tell you about the shape of the curve \(y = g ( x )\) as \(x \rightarrow - \infty\) ? [0pt]
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\section*{4. In this question you must show detailed reasoning}
The non-zero coefficients of \(x , x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 1 + x ) ^ { n }\) form the first, second and third terms of an arithmetic sequence (in that order).
Determine the possible value(s) of \(n\).
For the same value(s) of \(n\), there is another value of \(a\) for which \(( 1 + a x ) ^ { n }\) has this property. Determine this value of \(a\).