Final answer:
The maximum rate of change of a function f(x, y) is found by calculating the magnitude of the gradient vector, which is composed of the function's partial derivatives with respect to x and y. The magnitude is computed as the square root of the sum of the squares of these partial derivatives.
Step-by-step explanation:
To find the maximum rate of change of a function f(x, y), you need to calculate the gradient of f at a particular point. In two dimensions, the rate of change in the direction of the steepest ascent is represented by the gradient vector, often denoted as ∇f or grad f. The magnitude of this gradient vector gives you the maximum rate of change of the function at that point.
The gradient vector is composed of the partial derivatives of f with respect to each variable. For a function f(x, y), the gradient is ∇f = (fx, fy), where fx is the partial derivative of f with respect to x, and fy is the partial derivative with respect to y. The maximum rate of change is thus the magnitude of the gradient vector, which is calculated as √(fx² + fy²). To find the actual maximum value, you might have to evaluate this magnitude at critical points determined by setting the partial derivatives equal to zero and solving for x and y.