Function UK MAT Intermediate

2007

Problem - 4758

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 $$\underbrace{f_2(f_2(\cdots f_2(x)))}_{k}$$

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