Final answer:
The maximum rate of change of f(x, y, z) = x / (x² y²) is given by |∂f/∂x|, which simplifies to 1/(x^3 y^2).
Step-by-step explanation:
The rate of change of a function can be found by taking the partial derivatives of the function with respect to each variable and then evaluating them at a given point. In this case, let's find the partial derivatives of f(x, y, z) = x / (x² y²):
- ∂f/∂x = (x² y² - 2xy²)/(x^4 y^4)
- ∂f/∂y = -2x/(x^4 y^3)
- ∂f/∂z = 0
To find the maximum rate of change, we need to find the maximum value of |∂f/∂x|, |∂f/∂y|, and |∂f/∂z|. The maximum rate of change is given by the largest of these values. From the derivatives we calculated, the largest value is |∂f/∂x| which simplifies to 1/(x^3 y^2). Therefore, the correct answer is (c) 1 / (x² y³).