Minimizing more than one variable

Algebra Level pending

A Function \(f(z)=(4+i){ z }^{ 2 }+\alpha z+\gamma \), (\(\alpha ,\gamma \in C\)) is such that both \(f(1)\) and \(f(i)\) are pure real quantities. Let the minimum value of \(\left| \alpha \right| +\left| \gamma \right| =k\) , Compute \(\left[ \frac { { k }^{ 2 } }{ 2 } \right] \) (\(\left[ . \right] \) denotes GIF).

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