Let $f_n (x) = (2 + (−2)^n ) x^2 + (n + 3) x + n^2$.
- Write down $f_3(x)$ and find its maximum value. Also determine for what value of $n$ does the function $f_n(x)$ have a maximum value (as $x$ varies). You do not need to compute this maximum value.
- Write down $f_1(x)$. Calculate $f_1(f_1(x))$ and $f_1(f_1(f_1(x)))$. Find an expression, simplified as much as possible, for $$\underbrace{f_1(f_1(\cdots f_1(x)))}_{k}$$
- Write down $f_2(x)$. Find the degree of the function $$\underbrace{f_2(f_2(\cdots f_2(x)))}_{k}$$
(1) Setting $n=3$ leads $f_3(x)=-6x^2 + 6x+ 9$. This function has a maximum value of ${\frac{21}{2}}$ when $x=\frac{1}{2}$. Furthermore, $f_n(x)$ can be either a quadratic polynomial of $x$ or a linear polynomial. It will have a maximum value if and only if it's quadratic and the leading coefficient is negative, i.e. $2+ (-2)^n < 0$. This condition is equivalent to $n$ is an odd and greater than $1$.
(2) Setting $n=1$ leads to $f_1(x) = 4x + 1$, $f_1(f_1(x)) = 4(4x + 1) + 1=16x + 5$ and $f_1(f_1(f_1(x)))=4(16x + 5)+1 = 64x + 21$. More generally, $$\underbrace{f_1(f_1(\cdots f_1(x)))}_{k}=4^kx + (1+4+4^2+\cdots +4^{k-1}) = 4^kx + \frac{4^k-1}{3}$$
(3) Setting $n=2$ leads to $f_2(x)=6x^2 + 5x + 4$. If follows that the degree of $$\underbrace{f_2(f_2(\cdots f_2(x)))}_{k}$$ must be $2^k$.