Question 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 $|\alpha| + |\gamma|?$