Final answer:
To find the equation of the tangent plane to the given surface at the given point, use partial derivatives to find the values of f_x and f_y. Then, substitute these values into the equation of the tangent plane to get the equation of the plane.
Step-by-step explanation:
To find the equation of the tangent plane to the surface at the given point, we can use the concept of partial derivatives. The equation of the tangent plane is given by: z - f(a,b) = fx(a,b)(x-a) + fy(a,b)(y-b).
For the given surface, f(x, y) = x^2 - 2xy + y^2 and the point is (3,5,4), so a = 3 and b = 5. Taking partial derivatives with respect to x and y, we get fx(x, y) = 2x - 2y and fy(x, y) = 2y - 2x.
Substituting these values into the equation of the tangent plane, we have: z - 4 = (2(3) - 2(5))(x-3) + (2(5) - 2(3))(y-5).