Final Answer:
The maximum rate of change of the function at the point (2,1) can be found by evaluating the magnitude of the gradient vector at that point.
Explanation:
To find the maximum rate of change of the function at a given point, we consider the gradient vector ∇f(x, y) at that point. The gradient vector gives the direction of the maximum rate of change, and its magnitude represents the maximum rate of change itself.
Let's suppose the function is f(x, y). At the point (2,1), we evaluate the gradient vector ∇f(2, 1). If the function is given explicitly, we compute the partial derivatives ∂f/∂x and ∂f/∂y. Then, the gradient vector is (∂f/∂x, ∂f/∂y).
Next, we substitute the point (2,1) into these partial derivatives to obtain the gradient vector ∇f(2, 1). After finding the components of the gradient vector, we calculate its magnitude using the formula |∇f(2, 1)| = √((∂f/∂x)² + (∂f/∂y)²). This magnitude represents the maximum rate of change of the function at the point (2,1).
By computing the gradient vector's magnitude at (2,1), we determine the maximum rate of change of the function at that specific point. This method enables us to identify the direction and magnitude of the steepest ascent or descent of the function from that point.