AMC 12 2008b Problem 19

A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$?

(Solution)

\textbf{(A)} 1 \textbf{(B)} \sqrt {2} \textbf{(C)} 2 \textbf{(D)} 2 \sqrt {2} \textbf{(E)} 4