Final answer:
To find all points at which the direction of fastest change of the function f(x, y) = -x² + y² - 2x + 4y is (1, 1), we need to find the gradient vector of the function and set it equal to (1, 1).
Step-by-step explanation:
To find all points at which the direction of fastest change of the function f(x, y) = -x² + y² - 2x + 4y is (1, 1), we need to find the gradient vector of the function and set it equal to (1, 1). The gradient vector of f(x, y) is given by ∇f(x, y) = (-2x - 2, 2y + 4). When we set this equal to (1, 1), we get the equations -2x - 2 = 1 and 2y + 4 = 1. Solving these equations will give us the points at which the direction of fastest change of the function is (1, 1).