The maximum rate of change of the function f(x, y) = 5 sin(xy) at the point (0,6) is 30, occurring in the direction of the vector 30i, which is along the x-axis.
To find the maximum rate of change of a function f(x, y) = 5 sin(xy) at a given point (0,6), we first need to compute the gradient vector of f, represented as ∇f or grad f. The gradient at any point gives the direction of the steepest ascent from that point; its magnitude gives the maximum rate of change at that point.
The gradient of f is given by taking the partial derivatives of f with respect to x and y:
- ∂f/∂x = 5y cos(xy)
- ∂f/∂y = 5x cos(xy)
Evaluating these partial derivatives at the point (0, 6) gives us:
- ∂f/∂x (0, 6) = 5*6 cos(0*6) = 30
- ∂f/∂y (0, 6) = 5*0 cos(0*6) = 0
The gradient vector at (0, 6) is then ∇f(0, 6) = 30i + 0j. The magnitude of this vector is the maximum rate of change at the point (0, 6), which is simply 30. The direction of maximum increase is along the vector 30i or along the x-axis.