Final answer:
The linearization of the function f(x,y,z) at a given point is the sum of the function's value at that point and the dot product of the gradient vector at that point with the vector (x-x₀, y-y₀, z-z₀).
Step-by-step explanation:
The linearization L(x,y,z) of the function f(x,y,z) = 6xy + 7yz + xz at a point (x₀, y₀, z₀) is found by taking the function's value at that point plus the dot product of the gradient of the function at that point with the vector (x-x₀, y-y₀, z-z₀). The gradient of f consists of the partial derivatives with respect to each variable:
- ∂f/∂x = 6y + z
- ∂f/∂y = 6x + 7z
- ∂f/∂z = 7y + x
For instance, at point a. (1,1,1), the gradient is (6(1) + 1, 6(1) + 7(1), 7(1) + 1) = (7, 13, 8). The linearization at (1,1,1) is then:
L(x,y,z) = f(1,1,1) + 7(x-1) + 13(y-1) + 8(z-1)
L(x,y,z) = 6(1)(1) + 7(1)(1) + (1)(1) + 7(x-1) + 13(y-1) + 8(z-1)
L(x,y,z) = 6 + 7 + 1 + 7x - 7 + 13y - 13 + 8z - 8
L(x,y,z) = 7x + 13y + 8z - 13
For the point b. (1,0,0) and c. (0,0,0), the process is similar but considering the values of x, y, and z at those points.