Final answer:
To linearize the function f(x,y) = x² + y² - xy at the point (-1,1), we calculate the partial derivatives, evaluate them at (-1,1), and use them along with the function value at that point to construct the linearization formula. This results in the linearization L = -3x + 3y - 3.
Step-by-step explanation:
Given the function: f(x,y) = x² + y² - xy at the point (-1,1), we aim to find the linearization of this function at that point. Linearization is a method to approximate the function near the point of interest using the first-order Taylor expansion. To do this, we first need to calculate the partial derivatives of f with respect to x and y, evaluate them at the point (-1,1), and then use them to construct the linearization formula.
The partial derivative with respect to x is fx(x,y) = 2x - y and with respect to y is fy(x,y) = 2y - x. Evaluating these at the point (-1,1) gives fx(-1,1) = -2 - 1 = -3 and fy(-1,1) = 2(1) - (-1) = 3.
The function value at the point is f(-1,1) = (-1)² + (1)² - (-1)(1) = 1 + 1 + 1 = 3. Therefore, the linearization can be written as L = fx(-1,1)(x - (-1)) + fy(-1,1)(y - 1) + f(-1,1). Substituting the values we calculated, we get L = -3(x + 1) + 3(y - 1) + 3.
After simplifying the expression, we have L = -3x + 3y - 3. So, if the linearization of the function is required in the form of "L = ax + by + c", then LL = -3x + 3y - 3.