Let \[f_n(x)=(2+(-2)^n)x^2+(n+3)x+n^2\] where \(n\) is a positive integer and \(x\) is any real number.

  1. 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)?

The green curve here is \(y=(2+(-2)^n)x^2+(n+3)x+n^2\), where \(n\) is a positive integer.

What happens as we vary \(n\)? Make sure you can explain algebraically the behaviour you can see here.