Final answer:
The rate of change of the function f(x, y, z) = x²y - xz is most rapid in the direction of its gradient. The gradient at the point (1, 1, 1) is (1, 1, -1), which corresponds to option C. Hence, (1, 1, -1) is the vector in the direction of the most rapid rate of change.
Step-by-step explanation:
The rate of change of a function in a particular direction is given by the gradient of the function dotted with the unit vector in that direction. The gradient of the function f(x, y, z) = x²y - xz at the point (1, 1, 1) can be found by taking the partial derivatives with respect to x, y, and z:
- ∂f/∂x = 2xy - z
- ∂f/∂y = x²
- ∂f/∂z = -x
At the point (1, 1, 1), the gradient vector (∇f) is (2(1)(1) - 1, 1², -1) = (1, 1, -1).
The direction of the most rapid increase is in the direction of the gradient, so the unit vector that has the same direction as the gradient is the direction of most rapid increase. Comparing this with the options given, option C. (1, 1, -1) is already a unit vector and matches the gradient's direction at the specified point. Thus, the rate of change of the function f(x, y, z) is most rapid in the direction of the vector (1, 1, -1).