Problem:
A function f is defined by f(z)=(4+i)z2+Ξ±z+Ξ³ for all complex numbers z, where Ξ± and Ξ³ are complex numbers and i2=β1. Suppose that f(1) and f(i) are both real. What is the smallest possible value of β£Ξ±β£+β£Ξ³β£ ?
Answer Choices:
A. 1
B. 2β
C. 2
D. 22β
E. 4 Solution:
Let Ξ±=a+bi and Ξ³=c+di, where a,b,c, and d are real numbers. Then f(1)=(4+a+c)+(1+b+d)i, and f(i)=(β4βb+c)+(β1+a+d)i. Because both f(1) and f(i) are real, it follows that a=1βd and b=β1βd. Thus