Final answer:
The direction of maximum rate of change for the function f(x,y)=2xsin(xy) at the point (2,3) can be found by calculating the gradient vector.
Step-by-step explanation:
The direction of maximum rate of change for a function can be found by calculating the gradient vector. In this case, we have the function f(x,y) = 2xsin(xy).
First, we find the partial derivatives of f with respect to x and y:
∂f/∂x = 2sin(xy) + 2xycos(xy)
∂f/∂y = 2x²cos(xy)
Next, we evaluate these partial derivatives at the point (2,3):
∂f/∂x(2,3) = 2sin(6) + 12cos(6) ≈ 10.88
∂f/∂y(2,3) = 8cos(6) ≈ 7.95
Thus, the maximum rate of change occurs in the direction given by the gradient vector, which is approximately (10.88, 7.95).