A function f is defined by $f(z) = (4 + i)z^2 + az + b$ for all complex numbers $z$, where $a$ and $b$ are complex numbers.If $f(1)$ and $f(i)$ are both real, then what is the smallest possible value of $|a| + |b|$ ?
I tried using the property $z = \bar{z}$ (conjugate) for $z=1$ and $z=i $ and then $|a + b|\le |a| + |b|$ but I did not got the answer.
Given that $f(1)$ and $f(i)$ are both real, you know that $$4+i+a+b\qquad\text{ and }\qquad -4-i+ai+b,$$ are both real, so $\operatorname{im}a+\operatorname{im}b=-1$ and $\operatorname{re}a+\operatorname{im}b=1$. Then what is the smallest value of \begin{eqnarray*} |a|+|b| &=&\sqrt{(\operatorname{re}a)^2+(\operatorname{im}a)^2} +\sqrt{(\operatorname{re}b)^2+(\operatorname{im}b)^2}\\ &=&\sqrt{(1-\operatorname{im}b)^2+(-1-\operatorname{im}b)^2} +\sqrt{(\operatorname{re}b)^2+(\operatorname{im}b)^2}. \end{eqnarray*}