Final answer:
To find the maximum rate of change of f(x,y) = 4sin(xy) at the point (0,6) and the direction in which it occurs, we calculate the partial derivatives and evaluate them at the given point. The maximum rate of change is 24, and it occurs in the direction of the vector (0,24).
Step-by-step explanation:
To find the maximum rate of change of f(x,y) = 4sin(xy) at the point (0,6) and the direction in which it occurs, we need to calculate the partial derivatives with respect to x and y, and then evaluate them at the given point. The partial derivative with respect to x is f'(x,y) = 4y*cos(xy), and the partial derivative with respect to y is f'(x,y) = 4x*cos(xy). Evaluating these derivatives at (0,6), we get f'(0,6) = 24*cos(0) = 24.
So the maximum rate of change is 24. In order to determine the direction, we need to find the gradient of the function at the given point. The gradient is a vector that points in the direction of steepest ascent. The gradient of f(x,y) is given by (∂f/∂x, ∂f/∂y). Evaluating the gradient at (0,6), we get the vector (0, 4*6*cos(0)) = (0,24).
Therefore, the maximum rate of change of f(x,y) = 4sin(xy) at the point (0,6) is 24, and it occurs in the direction of the vector (0,24).