Final answer:
The equation of the tangent plane to f(x, y) = x² - 2xy + 2y² can be found using partial derivatives. The tangent plane equation is given by f(x₀, y₀) + (∂f/∂x)(x - x₀) + (∂f/∂y)(y - y₀) = 0, where (x₀, y₀) is a point on the function's surface.
Step-by-step explanation:
The equation of the tangent plane to the function f(x, y) = x² - 2xy + 2y² can be found using partial derivatives. To find the equation of the tangent plane, we need the gradient vector of the function. The gradient vector is given by (∂f/∂x, ∂f/∂y). In this case, the gradient vector is (2x - 2y, -2x + 4y). At a point (x₀, y₀) on the surface of the function, the equation of the tangent plane is given by:
f(x₀, y₀) + (∂f/∂x)(x - x₀) + (∂f/∂y)(y - y₀) = 0
Substituting the values from our function, we have:
(x₀)² - 2(x₀)(y₀) + 2(y₀)² + (2x₀ - 2y₀)(x - x₀) + (-2x₀ + 4y₀)(y - y₀) = 0
This equation represents the tangent plane to the given function.