Problem:
A function f is defined by f(z)=(4+i)z2+αz+γ for all complex numbers z, where α and γ are complex numbers and i2=−1. Suppose that f(1) and f(i) are both real. What is the smallest possible value of ∣α∣+∣γ∣ ?
Answer Choices:
A. 1
B. 2​
C. 2
D. 22​
E. 4 Solution:
Let α=a+bi and γ=c+di, where a,b,c, and d are real numbers. Then f(1)=(4+a+c)+(1+b+d)i, and f(i)=(−4−b+c)+(−1+a+d)i. Because both f(1) and f(i) are real, it follows that a=1−d and b=−1−d. Thus