jmc

Question

A function f is defined by f(z) = (4 + i) z^2 + z + for all complex numbers z, where and are complex numbers and i^2 = - 1. Suppose that f(1) and f(i) are both real. What is the smallest possible value of | | + | |?

Accepted Answer

Let = a + bi and = c + di, where a, b, c, and d are real numbers. Then align* f(1) &= (4 + i) + + = (a + c + 4) + (b + d + 1)i, f(i) &= (4 + i)(-1) + i + = (-b + c - 4) + (a + d - 1)i. align*Since f(1) and f(i) are both real, b + d + 1 = 0 and a + d - 1 = 0, so a = -d + 1 and b = -d - 1. Then align* || + || &= a^2 + b^2 + c^2 + d^2 &= (-d + 1)^2 + (-d - 1)^2 + c^2 + d^2 &= 2d^2 + 2 + c^2 + d^2 & 2. align*Equality occurs when a = 1, b = -1, c = 0, and d = 0. Therefore, the minimum value is 2.