\section*{2. For ALL APPLICANTS.}
Let

$$f _ { n } ( x ) = \left( 2 + ( - 2 ) ^ { n } \right) x ^ { 2 } + ( n + 3 ) x + n ^ { 2 }$$

where $n$ is a positive integer and $x$ is any real number.\\
(i) Write down $f _ { 3 } ( x )$.

Find the maximum value of $f _ { 3 } ( x )$.\\
For what values of $n$ does $f _ { n } ( x )$ have a maximum value (as $x$ varies)?\\[0pt]
[Note you are not being asked to calculate the value of this maximum.]\\
(ii) Write down $f _ { 1 } ( x )$.

Calculate $f _ { 1 } \left( f _ { 1 } ( x ) \right)$ and $f _ { 1 } \left( f _ { 1 } \left( f _ { 1 } ( x ) \right) \right)$.\\
Find an expression, simplified as much as possible, for

$$f _ { 1 } \left( f _ { 1 } \left( f _ { 1 } \left( \cdots f _ { 1 } ( x ) \right) \right) \right)$$

where $f _ { 1 }$ is applied $k$ times. [Here $k$ is a positive integer.]\\
(iii) Write down $f _ { 2 } ( x )$.

The function

$$f _ { 2 } \left( f _ { 2 } \left( f _ { 2 } \left( \cdots f _ { 2 } ( x ) \right) \right) \right) ,$$

where $f _ { 2 }$ is applied $k$ times, is a polynomial in $x$. What is the degree of this polynomial?