Consider the function f(y) = max(-y, y - 2). In which range does f take its minimum value?

A. -6

Question

Consider the function f(y) = max(-y, y - 2). In which range does f take its minimum value?

A. -6 <= y <= -2    B. -2 <= y <= 0    C. 0 <= y <= 2    D. 0 <= d <= 10

Accepted Answer

f(y) attains its minimum value of -1 at y = 1, which lies in the range 0 <= y <= 2 (option C). The function f(y) = max(-y, y-2) returns the larger of two linear expressions at each y. The function -y is a decreasing line and y-2 is an increasing line. They cross where -y = y-2, giving y = 1. For y < 1, -y > y-2 so f = -y; for y > 1, y-2 > -y so f = y-2. The minimum of f occurs exactly at the crossing point where both lines have the same value: f(1) = -1. The minimum value -1 is achieved only at y = 1, which lies in the range 0 <= y <= 2.