Final answer:
The linearization of the function f(x,y,z) = 8xy + yz + 5xz at the point (1, 1, 1) is obtained by evaluating the original function and its gradient at the point and then applying the dot product with the vector (x-1, y-1, z-1). The linearized function L(x, y, z) is then 14 + 13(x-1) + 9(y-1) + 6(z-1).
Step-by-step explanation:
To find the linearization L(x,y,z) of the function f(x,y,z) = 8xy + yz + 5xz at the point (1, 1, 1), we first need to compute the gradient of the function at the given point. The gradient is the vector of all first partial derivatives of f with respect to x, y, and z.
The partial derivative of f with respect to x at the point (1,1,1) is:
fₓ(1,1,1) = 8y + 5z = 8(1) + 5(1) = 13
The partial derivative of f with respect to y at the point (1,1,1) is:
fₒ(1,1,1) = 8x + z = 8(1) + 1(1) = 9
The partial derivative of f with respect to z at the point (1,1,1) is:
fₛ(1,1,1) = y + 5x = 1(1) + 5(1) = 6
The gradient at point (1, 1, 1) is thus (13, 9, 6). Now, the linearization is given by the function itself evaluated at the point plus the dot product of the gradient and the vector (x-1, y-1, z-1).
The value of f at point (1, 1, 1) is:
f(1, 1, 1) = 8(1)(1) + 1(1) + 5(1)(1) = 14
Therefore, the linearization L(x, y, z) is:
L(x, y, z) = f(1, 1, 1) + ➡⋅(x-1, y-1, z-1) = 14 + 13(x-1) + 9(y-1) + 6(z-1)