If $f(1)$ and $f(i)$ real, then find minimum value of $|a|+|b|$ A function $f$ is defined by $$f(z)=(4+i)z^2+a  z+b$$ $(i=\sqrt{-1})$for all complex number $z$ where $a$ and $b$ are complex numbers. If $f(1)$ and $f(i)$ are both purely real, then find minimum value of $|a|+|b|$
Now
$f(1)=4+i+a+b$ which means imaginary part of $a+b=-1$ and $f(i)=-4-i+a \cdot i+b$ which gives imaginary part of $a \cdot i+b=1$  but how to proceed further to find desired value?

There is nothing to force the real part of $b$ to be other than zero and having it nonzero would make $|b|$ greater, so $b$ is purely imaginary.  Let $b=ci, a=a_r+a_ii$.  We know $c+a_i=-1, a_r+c=1$, so $|a|+|b|=\sqrt{(c+1)^2+(c-1)^2}+c$  Minimize this over $c$.
