Let f(x) = {x² - 4x + 3 for x < 2; ax + b for x ≥ 2}. For what values of a and b is f both continuous and differentiable at x = 2?

Core Concept

For continuity at x = 2: lim(x→2⁻)f(x) = 4 - 8 + 3 = -1, so f(2) must equal -1, giving 2a + b = -1. For differentiability: f'(x) from left is 2x - 4, so f'(2⁻) = 0; f'(x) from right is a, so a = 0. Then b = -1.